 Commutation of fixed points and orbit quotients for coprime groups

For commuting actions of finite groups $G,H$ with coprime orders, the canonical map $A^G/H\to (A/H{)}^G$ is a bijection. It is always injective. On any $G$-stable $H$-orbit, all $G$-orbits have the same size, dividing both $|G|$ and the $H$-orbit size, hence dividing $|H|$. Coprimality forces that size to be one, proving surjectivity. Thus categorical limits indexed by $G$ commute with colimits indexed by $H$ in the Category of sets.