Gittin 52 - Chinuch by an Apitropes

When one adopts a Jewish child (or ger kattan), is there an actual obligation of chinuch incumbent upon the parents who adopted the child? It seems clear from the gemara that there is. The gemara says that the caretaker of the orphans has the authority (and requirement) to purchase for them the articles with which they can fulfill positive mitzvos such as lulav, sucah tzitzis, tefillin and mezuza e.t.c. Why should he use their money to purchase these things, since they are children and exempt from mitzvos? Shulchan Aruch (C.M. 290:15) writes that although they are exempt there is indeed a mitzvah of chinuch incumbent on the apitropes. The Aruch Hashulchan explains a little more clearly that the caretaker is literally in place of the father and takes over all of his obligations. Based on this it seems clear that a father who adopted a child would be no less than a caretaker (especially since the mishnah indicates that a caretaker need not be appointed, so long as he is the person that the children rely on). It would follow that if a single woman would adopt a child, she too would have status of the "caretaker" and would be obligated to provide the child with these mitzvah articles for chinuch purposes (although the beis din would not appoint a woman, rashi explains that is only to deal with their money for business purposes, but if she was appointed by the father she would have the full jurisdiction of an apirtropes). This seems to contradict the discussion of the achronim (see magen avrohom o.c. 343) who say that a mother is not obligated in the mitzvah of chinuch for a child. Perhaps this should be true in the presence of a father who can do it, but in the absence of a father, the mother should be no worse that an apitropes who would be obligated in their chinuch. This distinction would answer r' akiva eiger's question in succah 3b, why hilni hamalka was obligated in chinuch for her sons - perhaps there was not father so it was incumbent upon her as an apitropes. However, it could be that the mitzvah of chinuch of the father would require him to use his own money to provide his sons with mitzvah articles, but the obligation on an apitropes would be to spend their money, not his own. Therefore, a mother (even the absence of a father) or a parent who adopted a child, may only be responsible as an apitropes, and therefore she would not be obligated to spend their own money.