Is the husband's taking back of his wife, whom he divorced with a single revocable divorce, by having intercourse with her during her waiting period (iddah) considered Islamically valid, given his refusal to inform her family and his recanting of the decision to take her back after the waiting period ended?

Verbal agreement and cohabitation are considered a 'Raja'a' (revocation of divorce). If some friends are aware of this from the husband, then the witnessing (Ishhad) has been achieved. What the husband has done by maintaining intimate relations while the wife continues to live with her family is 'Imsak bi ghayri ma'ruf' (holding on in an unbefitting manner) and is harmful to her. The husband must either 'Imsak bi ma'ruf' (hold on in a befitting manner) or 'Tasrih bi ihsan' (release with kindness). The husband must announce his wife's return and have friends witness it; otherwise, he must divorce her. It is not permissible for him to leave her in this harmful situation.