Is it permissible for a husband to take back his wife after she committed adultery and eloped with a foreign man, given that the husband knows of her truthfulness that she was not touched, and his family rejects this?

The woman's actions of traveling and having a relationship with a foreign man are a sin, an act of disobedience, a betrayal of her husband, and a defilement of her honor. She must repent. If she repents sincerely, then it is permissible for her husband to keep her, provided he ensures she is not pregnant by observing the waiting period ('iddah), because her claim of not having intercourse cannot be accepted as truth. Obedience to parents is obligatory in what is good and permissible, and divorcing a wife who has repented sincerely is not considered good. However, if she does not repent, then divorcing her becomes necessary to satisfy the mother, protect the marital bed, and safeguard the lineage.