Is it obligatory for a girl to sacrifice her studies abroad and stay with her mother, who fears loneliness during the father's absence for work, knowing that staying in her current country will not enable her to obtain a good specialization?

It is best to combine good specialization with the mother's satisfaction, by the father providing a maid for the mother if the matter is merely her unwillingness to stay alone. However, if she wishes her daughter to remain with her, it is better to be dutiful to her and search for a possible study in the same country or remotely. It is permissible for a son to travel for seeking knowledge or trade without the permission of his parents if he does not fear for their neglect and is safe for himself. A woman's travel requires a mahram, but a mahram is not required for staying with her, rather she must be in a place where she feels safe.