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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.

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    $\begingroup$ Presumably you mean group structures on $H$ such that there is a group embedding $i : G \to H$ over $G$ (or maybe I mean under …), together with some hypothesis on the resulting projection $H/i(G) \to X$ (maybe that the projection intertwines the action of $G$ on $H/i(G)$ by left translation with the given action of $G$ on $X$)? $\endgroup$
    – LSpice
    Commented Jul 2, 2021 at 16:42
  • $\begingroup$ In theory cotriple cohomology provides an answer for all extension problems of this sort; in practice, it's seldom computable. $\endgroup$
    – Denis T
    Commented Jul 3, 2021 at 18:09
  • $\begingroup$ @DenisT. can you say exactly which of cohomology group control this extension problem? Meaning specify the cotriple, the functors, and where one has to compute them. Thanks! $\endgroup$
    – Giulio
    Commented Jul 4, 2021 at 21:15

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