Common quotient obstruction to commutation of fixed points and orbits
= Common quotient obstruction to commutation of fixed points and orbits
If <groups> $G,H$ have a common nontrivial quotient $K$, act on $K$ by left multiplication through $G$ and right inverse multiplication through $H$. The actions commute. There are no <fixed points of a group action> for $G$, but the $H$ orbit set is a singleton. The comparison $A^G/H\to(A/H)^G$ is therefore the map from the empty set to a singleton, which is not an <isomorphism>.