Does divorce occur if the mother gives money to her sister (the son's aunt) with the intention of paying off her spendthrift son's debt, after her husband swore an oath of divorce upon her not to give him money?

If the reason your father swore by divorce to prevent giving his son money was extravagance, then whether or not the oath is broken depends on whether paying off his debt falls under extravagance. If paying off the debt encourages the son to incur more debt and continue his extravagance, then the oath is broken; otherwise, it is not. If the situation is unknown, it is better to refrain from paying off the debt. The view of many scholars and our preponderant opinion is that divorce occurs in this case, whereas Sheikh Al-Islam Ibn Taymiyyah holds that an expiation for an oath is due if the father did not intend divorce. It is not permissible for the mother to pay off the son's debt from the father's money without his permission, because it is disposing of another's property without their permission, which is forbidden. Likewise, swearing by divorce is impermissible.