What is the ruling on returning a divorced woman after an innovated divorce (طلاق بدعة) without a new marriage contract, and is her return considered disobedience to her mother if the mother does not wish her to return?

Divorce pronounced during menstruation or during a period of purity in which intercourse occurred is a valid divorce, despite being an innovation (bid'ah). This is the view of the majority of scholars.

If the divorce does not complete the three divorces, it is permissible for the husband to take his wife back during her iddah (waiting period) without a new marriage contract. Her iddah is three menstrual cycles, or three months, or by giving birth for a pregnant woman.

If the wife has experienced three menstrual cycles after the divorce, she becomes irrevocably separated, and he cannot take her back except with a new contract and a new mahr (dowry). If her menstruation stops due to breastfeeding, her iddah continues until her menstruation returns and she experiences three menstrual cycles.

If the wife is righteous, you are not obliged to obey your mother in divorcing her or not taking her back. This is not considered disobedience to your mother. Ibn Taymiyyah was asked about a married man whose mother disliked his wife and advised him to divorce her. He replied: "It is not permissible for him to divorce her because of his mother's words. Rather, he must honor his mother, and divorcing his wife is not part of honoring her."

You should honor your mother and be good to her, and strive to reconcile between her and your sister's husband.