Fasting on behalf of someone else?
A person cannot observe the missed (Qaza) fasts of another person on their behalf; rather, the one upon whom the fast is obligatory is strictly required to fulfill it. However, if a person becomes so incapacitated from fasting due to an illness or old age that there is no hope of them regaining the ability to fast until death, then such a person can give Fidya (expiation) in exchange for the fast, the amount of which is equal to one Sadaqah al-Fitr. But their obligatory fast will not be fulfilled if another person fasts in their place. (Book: Kitab an-Nawazil / Vol: 6 / Pg: 398)