If a Muslim swears an oath to a promise, will he be held accountable for his inner intention or for what the other party understood, given that she knows her intention and what the other expects? And what is the expiation for that?

Your oath is based on what your interlocutor believes you are swearing about. This rule is confirmed if the intention of the oath is to invalidate another's right. Therefore, the questioner's intention to specify the oath for one friend is not valid. Rather, her oath encompasses her mother's intention concerning this entire group. Thus, she is obligated to leave this whole group. If she fails to do so, she must offer expiation for breaking an oath by feeding ten poor people, clothing them, or fasting for three days if she is unable to do the former. It is obligatory to completely abandon bad company due to its serious effect on a person, for a man is upon the religion of his intimate friend, and you should only associate with a believer.