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 $Rep(H)$).

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 $H-$equivariant objects of $Rep(K)$).