Is it permissible for a Muslim woman to travel and leave her family if she is psychologically exhausted from living abroad and her father refuses her return?

What is advised is to obey the father and stay with him, as long as no harm to one's religion or worldly life results from it. The lack of friends is a minor issue and can be remedied by seeking good company. Obeying the father in what is good is a cause for success, prosperity, and entering Paradise, based on the Prophet's (peace be upon him) saying: "The father is the middle gate of Paradise; so, if you wish, squander this gate or preserve it." However, if there is actual harm in staying with the father, then obedience to him in that specific matter is not obligatory. In all cases, one must be dutiful and kind to the father, and not harm him or abandon him.