If a God in his/her incarnation commits a sin is he/she ever punished? Or are they exempted from punishment because they are Gods?
Examples will help.
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If you see the below text you will see that Indra/Vishnu who are God were also afflicted by sin, so do you think an Avataar if he/she commits a sin would be spared?
SB 6.13: King Indra Afflicted by Sinful Reaction
This chapter describes Indra’s fear at having killed a brāhmaṇa (Vṛtrāsura), and it also describes how he fled and was saved by the grace of Lord Viṣṇu.
When all the demigods requested Indra to kill Vṛtrāsura, he refused because Vṛtrāsura was a brāhmaṇa. The demigods, however, encouraged Indra not to fear killing him because Indra was protected by the Nārāyaṇa-kavaca, or the Supreme Personality of Godhead Himself, Lord Nārāyaṇa. Even by a glimpse of the chanting of Nārāyaṇa’s name, one becomes free from all the sinful reactions of killing a woman, a cow or a brāhmaṇa. The demigods advised Indra to perform an aśvamedha sacrifice, by which Nārāyaṇa would be pleased, for the performer of such a sacrifice is not implicated in sinful reactions even if he kills the entire universe.
Also refer text from Garuda Puran that supports the above story:
The murderer of a Brāhmaṇ becomes consumptive, the killer of a cow becomes hump-backed and imbecile, the murderer of a virgin becomes leprous,--all three born as outcastes.
Curse of Jalandhar's wife
In another time, there is a great Rakshasa named Jalandhara. Shiva fights a fierce battle against him, but is unable to kill the demon. Vishnu learns that Jalandhara's immunity it due to the piousness of his wife, Vrinda. So, Vishnu takes the form of Jalandhara in front of Vrinda, and she embraces him thinking he is her husband. When Vrinda realizes she was tricked, she curses Vishnu that some day his own wife will be stolen from him, and then commits suicide. With Vrinda now deceased, Shiva is able to defeat Jalandhara.