What is the ruling on returning a greeting secretly, without uttering it aloud, if the one who offered the greeting was a woman to another woman, and there is avoidance in dealings between them?

Returning the greeting of Salam is obligatory by consensus. The obligation of returning Salam is not fulfilled by an inaudible reply, unless the person greeted is deaf, in which case the reply should be made by moving the lips.

If there is estrangement between two Muslim women, it is not permissible for it to exceed three days, based on the saying of the Prophet, peace and blessings be upon him: "It is not permissible for a believer to shun another believer for more than three days. If three days pass, he should meet him and greet him with Salam. If the other person returns the Salam, they both share in the reward. If he does not return it, then the sin is upon him, and the Muslim has exited the state of shunning."

However, if the one greeting is a transgressor (fasiq), it is permissible to return the Salam silently or not to return it at all if there is a benefit in doing so, as the Prophet, peace and blessings be upon him, did with Ka'b ibn Malik, may Allah be pleased with him.