Let $G$ be a finite group and $X$ a finite $G$-set. Let $H$ be the set-theoretical cartesian product of $G$ and $X$.

Is there an homological theory controlling all possible group structure on $H$ (possibly assuming that there exists at least one group structure) ?

I am also interested in answers with the extra assumption that $G$ is commutative, and then looking either for commutative group structures or all group structures on $H$.

Any reference is welcome.