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This is just a reiteration of David's answer: in this situation the group $H$ acts on the category $Rep(K)$ (an element $h\in H$ sends a representation of $K$ into its twist by $\phi(h)$; I should add that the notion of an action of affine group on abelian category is a bit subtle..). In this language, the category $Rep(G)$ is just an "equivariantization" of the category $Rep(K)$ (in other words the objects of $Rep(G)$ are just equivariant $H-$equivariant objects of $Rep(H)$).Rep(K)$). 1 This is just a reiteration of David's answer: in this situation the group$H$acts on the category$Rep(K)$(an element$h\in H$sends a representation of$K$into its twist by$\phi(h)$; I should add that the notion of an action of affine group on abelian category is a bit subtle..). In this language, the category$Rep(G)$is just an "equivariantization" of the category$Rep(K)$(in other words the objects of$Rep(G)$are just equivariant objects of$Rep(H)\$).