What is the religious ruling concerning a wife who refuses intimacy with her husband, speaks with foreign men online, and rejects her husband's instructions, and the question arises regarding the father's custody of his children from her should he wish to divorce her?

We advise you to prevent your wife from working in mixed environments and to teach her matters of her religion. If she does not respond after admonition, boycotting (emotional distance), and light, non-bruising striking, then divorce is the last resort. The custody of children belongs to the mother unless there is an impediment, then to their maternal grandmother. If the spouses separate and reside in different countries, the father has a greater right to custody according to most scholars. The final arbiter in the matter of dispute is the Sharia court.