The terms "pious" and "impious" are applicable only in reference to the activities of an ordinary living being. But Mahārāja Pṛthu was a directly empowered incarnation of Lord Viṣṇu; therefore he was not subject to the reactions of pious or impious activities. As we have already explained previously, when a living being is specifically empowered by the Supreme Lord to act for a particular purpose, he is called a śaktyāveśa-avatāra. Pṛthu Mahārāja was not only a śaktyāveśa-avatāra but also a great devotee. A devotee is not subjected to the reactions resulting from past deeds. In the Brahma-saṁhitā it is said, karmāṇi nirdahati kintu ca bhakti-bhājām: (Bs. 5.54) for devotees the results of past pious and impious activities are nullified by the Supreme Personality of Godhead.
(Srimad Bhagavatam----4:21:11----purport).

Yamarāja, the great controller of life after death, decides the living entities' destinies in their next lives. He is surely among the most confidential representatives of the Lord. Such confidential posts are offered to great devotees of the Lord who are as good as His eternal associates in the spiritual sky. And because Vidura happened to be among them, the Lord, while returning to Vaikuṇṭha, left instructions for Vidura with Maitreya Muni. Generally the eternal associates of the Lord in the spiritual sky do not come to the material world. Sometimes they come, however, by the order of the Lord—not to hold any administrative post, but to associate with the Lord in person or to propagate the message of God in human society. Such empowered representatives are called śaktyāveśa-avatāras, or incarnations invested with transcendental power of attorney.
(Srimad Bhagavatam----3:5:21----purport).

In Bhagavad-gītā the Lord says that whenever one sees an extraordinary power, he should conclude that a specific partial representation of the Supreme Personality of Godhead is present. There are innumerable such personalities, but not all of them are direct viṣṇu-tattva plenary expansions of the Lord. Many living entities are classified among the śakti-tattvas. Such incarnations, empowered for specific purposes, are known as śaktyāveśa-avatāras. King Pṛthu was such a śaktyāveśa-avatāra of the Lord. Similarly, Arci, King Pṛthu's wife, was aśaktyāveśa-avatāra of the goddess of fortune.
(Srimad Bhagavatam----4:15:6----purport).

Thus a learned man does not look upon the dresses that externally cover the living entity, but sees the pure soul within the varieties of dress and knows very well that the varieties of dress are the creation of nescience (avidyā-racitam). Being a śaktyāveśa-avatāra, empowered by the Supreme Personality of Godhead, Pṛthu Mahārāja did not change his spiritual position, and consequently there was no possibility of his viewing the material world as reality.
(Srimad Bhagavatam----4:16:19----purport).

Sometimes Lord Viṣṇu appears in His person as Lord Kṛṣṇa or Lord Rāma. All of these appearances are mentioned in the śāstras. Sometimes He appears as a śaktyāveśa-avatāra like Lord Buddha. As explained before, these śaktyāveśa-avatāras are incarnations of Viṣṇu's power invested in a living entity. Living entities are also part and parcel of Lord Viṣṇu, but they are not as powerful; therefore when a living entity descends as an incarnation of Viṣṇu, he is especially empowered by the Lord.
When King Pṛthu is described as an incarnation of Lord Viṣṇu, it should be understood that he is a śaktyāveśa-avatāra, part and parcel of Lord Viṣṇu, and is specifically empowered by Him. Any living being acting as the incarnation of Lord Viṣṇu is thus empowered by Lord Viṣṇu to preach the bhakti cult. Such a person can act like Lord Viṣṇu and defeat demons by arguments and preach the bhakti cult exactly according to the principles of śāstra. As indicated in Bhagavad-gītā, whenever we find someone extraordinary preaching the bhakti cult, we should know that he is especially empowered by Lord Viṣṇu, or Lord Kṛṣṇa. As confirmed in Caitanya-caritāmṛta (cc.3:7:11), kṛṣṇa-śakti vinā nahe tāra pravartana: one cannot explain the glories of the holy name of the Lord without being specifically empowered by Him. If one criticizes or finds fault with such an empowered personality, one is to be considered an offender against Lord Viṣṇu and is punishable.
(Srimad Bhagavatam----4:19:37----purport).

Since Lord Śiva is in charge of annihilation, why should he be approached for protection, which is given by Lord Viṣṇu? Lord Brahmā creates, and Lord Śiva annihilates, but both Lord Brahmā and Lord Śiva are incarnations of Lord Viṣṇu and are known as śaktyāveśa-avatāras. They are endowed with a special power like that of Lord Viṣṇu, who is actually all-pervading in their activities. Therefore whenever prayers for protection are offered to Lord Śiva, actually Lord Viṣṇu is indicated, for otherwise Lord Śiva is meant for destruction. Lord Śiva is one of the īśvaras, or the controllers known as śaktyāveśa-avatāras. Therefore he can be addressed as having the qualities of Lord Viṣṇu.
(Srimad Bhagavatam----8:7:21----purport).

Out of His causeless mercy and compassion, Kṛṣṇa has compiled various Vedic literatures in His incarnation as Vyāsadeva. Vyāsadeva is a śaktyāveśa-avatāra of Lord Kṛṣṇa. He has very kindly presented these literatures to awaken the conditioned soul to his senses. Unfortunately, at the present moment the conditioned souls are guided by demons who do not care to read the Vedic literatures. Although there is an immense treasure-house of knowledge, people are engaged in reading useless literature that will give them no information on how to get out of the clutches of māyā.
(Sri Caitanya Caritamrta----2:20:122----purport).

The śaktyāveśa-avatāras are categorized into (1) forms of divine absorption (bhagavad-āveśa), such as Kapiladeva or Ṛṣabhadeva, and (2) divinely empowered forms (śaktyāveśa), of whom seven are foremost: (1) Śeṣa Nāga in the Vaikuṇṭha world, empowered for the personal service of the Supreme Lord (sva-sevana-śakti), (2) Anantadeva, empowered to bear all the planets within the universe (bhū-dhāraṇa-śakti), (3) Lord Brahmā, empowered with the energy to create the cosmic manifestation (sṛṣṭi-śakti), (4) Catuḥsana, or the Kumāras, specifically empowered to distribute transcendental knowledge (jñāna-śakti), (5) Nārada Muni, empowered to distribute devotional service (bhakti-śakti), (6) Mahārāja Pṛthu, specifically empowered to rule and maintain the living entities (pālana-śakti) and (7) Paraśurāma, specifically empowered to cut down rogues and demons (duṣṭa-damana-śakti).
(Sri Caitanya Caritamrta----2:20:246----translation).

By devotional service one can understand that Kṛṣṇa first of all manifests Himself as svayam-rūpa, His personal form, then as tadekātma-rūpa, and then as āveśa-rūpa. It is in these three features that He manifests Himself in His transcendental form. The feature of svayam-rūpa is the form by which Kṛṣṇa can be understood by one who may not understand His other features. In other words, the form by which Kṛṣṇa is directly understood is called svayam-rūpa, or His personal form. The tadekātma-rūpa is that form which most resembles the svayam-rūpa, but there are some differences in the bodily features. The tadekātma-rūpa is divided into two manifestations—the personal expansion (svāṁśa) and the pastime expansion (vilāsa). As far as the āveśa-rūpa is concerned, when Kṛṣṇa empowers some suitable living entity to represent Him, that living entity is called āveśa-rūpa, orśaktyāveśa-avatāra.
(Teachings of Lord Caitanya).

The Lord then pointed out that there is no limit to the śaktyāveśa-avatāras and that they cannot be counted. However, some can be mentioned as examples. The śaktyāveśa incarnations are of two kinds—direct and indirect. When the Lord Himself comes, He is called sākṣāt, or a direct śaktyāveśa-avatāra, and when He empowers some living entity to represent Him that living entity is called an indirect or āveśa incarnation. Examples of indirect avatāras are the four Kumāras, Nārada, Pṛthu and Paraśurāma. These are actually living entities, but there is specific power given to them by the Supreme Personality of Godhead. When a specific opulence of the Supreme Lord is invested in specific entities, they are called āveśa-avatāras. The four Kumāras specifically represent the Supreme Lord's opulence of knowledge. Nārada represents the devotional service of the Supreme Lord. Devotional service is also represented by Lord Caitanya, who is considered to be the full representation of devotional service. In Brahmā the opulence of creative power is invested, and in King Pṛthu the power for maintaining the living entities is invested. Similarly, in Paraśurāma the power for killing evil elements is invested. As far as vibhāti, or the special favor of the Supreme Personality of Godhead, is concerned, it is described in the Tenth Chapter of Bhagavad-gītā that a living entity who appears to be especially powerful or beautiful should be known to be especially favored by the Supreme Lord.
(Teachings of Lord Caitanya).

The four yuga-avatāras are (1) śukla (white) in Satya-yuga (S.B.11:5:21), (2) rakta (red) in Tretā-yuga (S.B.11:5:24), (3) śyāma (dark blue) in Dvāpara-yuga (S.B.11:5:27) and (4) generally kṛṣṇa (black) but in special cases pīta (yellow) as Caitanya Mahāprabhu in Kali-yuga (S.B.11:5:32) and (S.B.10:8:13).
(Sri Caitanya Caritamrta----2:20:246----purport).

O Sanātana, now hear from Me about the yuga-avatāras, the incarnations for the millenniums. First of all, there are four yugas—Satya-yuga, Tretā-yuga, Dvāpara-yuga and Kali-yuga.
(Sri Caitanya Caritamrta----2:20:339----purport).

In Satya-yuga the Lord appeared in a body colored white, with four arms and matted hair. He wore tree bark and bore a black antelope skin. He wore a sacred thread and a garland of rudrākṣa beads. He carried a rod and a waterpot, and He was a brahmacārī.
(Sri Caitanya Caritamrta----2:20:332----purport).

In Tretā-yuga, the Lord appeared in a body that had a reddish hue and four arms. There were three distinctive lines on His abdomen, and His hair was golden. His form manifested the Vedic knowledge, and He bore the symbols of a sacrificial spoon, ladle and so on.
(Sri Caitanya Caritamrta----2:20:333----purport).

In Dvāpara-yuga the Personality of Godhead appears in a blackish hue. He is dressed in yellow, He holds His own weapons, and He is decorated with the Kaustubha jewel and the mark of Śrīvatsa. That is how His symptoms are described.
(Sri Caitanya Caritamrta----2:20:337----purport).

In the Age of Kali, Lord Kṛṣṇa assumes a golden color and, accompanied by His personal devotees, introduces hari-nāma-saṅkīrtana, the chanting of the Hare Kṛṣṇa mantra. By this process He delivers love for Kṛṣṇa to the general populace.
(Sri Caitanya Caritamrta----2:20:340----purport).