Is a wife in Egypt obligated to fulfill her husband's request to take his daughter to live with her in Egypt, knowing the daughter's desire to stay with her mother?

The default rule is that the custody of a child belongs to the mother, unless there is a legitimate religious impediment. Thus, it is not permissible for the father to take his daughter away from her mother's custody unless the mother is unfit for custody. In such a case, the matter should be referred to the Sharia court for a ruling. If the court grants custody to the father, then there is no harm in his other wife having custody of the girl. It should be noted that the mother's established custody does not prevent the father from seeing his daughter, caring for her, and disciplining her.