Is it permissible to continue a marriage with a wife who sent personal pictures to a stranger and received gifts from him without her husband's knowledge, despite her admission of the mistake and swearing on the Qur'an not to betray, while doubts persist about the nature of their relationship?

If what you have mentioned about your wife — that she sent her photos and her friend’s photos to a foreign man and received gifts from him — is true, then this is a reprehensible act for which she must offer sincere repentance, even if her intention was good. As long as your wife maintains her prayers, recites the Qur’an, and avoids prohibitions, then keep her and treat her well, and strive to nurture her in goodness and faith. You must also forget what happened and not dwell on ill-suspicion towards her, so that Satan does not exploit this to cause separation between you.