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Interactions (functors) between equivariant sheaves for different groups?

To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. … It is essentially obvious that all standard operations on sheaves (pull,push,shriek push,tensor etc.) elevate to the level of $G$-equivariant sheaves on $G$-sets and with $G$-equivariant maps. …