Is it permissible for my grandmother's sons to perform the sixty-day expiation on her behalf, after she unintentionally caused the death of her infant son, and how can they cooperate in performing this expiation?

The grandmother's act of killing her son is considered accidental killing. As for fasting on her behalf, there is a difference of opinion among scholars: some have permitted it absolutely, some have forbidden it absolutely, and some have restricted it to vows. Imam An-Nawawi and Ash-Shawkani preferred the permissibility of a guardian fasting on behalf of the deceased absolutely. The most cautious approach is for one family member to fast on her behalf, with the requirement that the one fasting be a single individual due to the necessity of continuous fasting.