As far as I can remember it from other answers I have seen on AB,the Mother needs the Father's permission for even a holiday.So taking the child to live abroad could be even more difficult.
Of course it all depends on what the legal custody arrangements are.
There ARE people on AB that have far more knowledge of such matters than me,but I just thought I would put a "stop gap" answer in until the morning when you may get more correct responses.
Jo.