Is it permissible to fast on behalf of a mother who experiences hardship in fasting, in order to atone for an oath she forgot and then began to command things contrary to its content, and does the phrase "I adjure you" count as an oath?

The jurists have differed on whether the phrase "I adjure you" (أقسمت عليك) constitutes an oath or not. Some, like Ibn Umar, Ibn Abbas, Al-Nakha'i, Al-Thawri, and the Kufans, said it is an oath absolutely. Others, the majority, said it is not an oath unless intended as such. Malik distinguished between "I adjure you by Allah" (أقسمت بالله), which is an oath, and "I adjure you" (أقسمت) in isolation, which is not an oath unless intended. Al-Shafi'i stated that the isolated phrase is not an oath at all, even with intention. Ishaq said it is not an oath at all. Ahmad has various narrations, similar to the first and second opinions, and a narration stating it is definitively an oath if one says "an oath by Allah" (قسمًا بالله).

Accordingly, the isolated phrase "I adjure you" (أقسمت عليك) is not an oath unless intended, according to the majority and Malik. The more probable view is that it is an oath if intended. Therefore, if your mother intended it as an oath and you disobeyed her, she owes an expiation for breaking an oath.

The expiation for an oath does not begin with fasting; rather, it begins with freeing a slave, feeding the poor, or clothing the poor. If these are not possible, then fasting three days. It is not permissible for you to fast on her behalf, as there is a consensus that no one fasts on behalf of a living person.

If your mother commands you to do something that contradicts her oath, inform her of this and give her the choice between breaking her oath or keeping it. Then, do what she decides.