Is it permissible to steal money to fend off a threat of false defamation? What is the ruling of Islamic law on false accusation? And how can a dispute between relatives be resolved due to this situation?

This answer conveys three messages:

The first is to the questioner: We ask Allah to magnify your reward and relieve your distress. It would have been better to consult people of knowledge before the problem escalated. What you did towards your aunt, in terms of honesty and restoring what is due, is your duty. We urge you to report your friend to the police to reveal the truth.

The second is to the questioner's friend: You must repent to Allah and seek forgiveness for what you did: betrayal, false accusation, spreading evil, and taking money unlawfully. The completion of your repentance involves admitting your mistake, returning the money, and destroying the picture.

The third is to the questioner's aunt: You are a refuge in times of difficulty. You must believe your niece because she swore by Allah. Whoever has an oath sworn to him by Allah should be content. Do not rule anything out when it comes to people of evil. Accept her oath and think well of her. If there was a sin, it is obligatory to conceal it, for whoever conceals a Muslim, Allah will conceal him.