I would like to ask for possible references for the following very general situation, a categorified version of Mackey functors.
The question is if there are other known constructions to associate to any soubgroup H of G a category C(H) and for any $H\leq K \leq G$ pairs of adjoint functors $Ind_H^K:C(H)\rightarrow C(K)$, $Res_H^K:C(K)\rightarrow C(H)$ satisfying analogues axioms to those from group theory ( i.e., when $C(H)=Rep(H)$)?
There are some very nice papers by Ocha & all considering Clifford theory for Mackey functors. Next natural question is can one define vertices and surces for these categorified Mackey functors? What about Green' s theorem in this categorified version from the previous question, are there other known examples?