Invariants and semi-direct products for finite group actions on tensor categories

Suppose a group G acts on a tenser category G over a field k. Then we have the tenser category G(G) of G-invariant objects in G, and the semi-direct product tenser category G[G]. We show that if G is finite and k[G] is semi-simple, there exists a one-to-one correspondence between categories with action of G(G) and categories with action of G[G].