*This is a follow up question to this question which remained unanswered (satisfactorily) even after a large bounty. I have made a litlle progress and I have no a more specific question which might be easier to answer. I hope that in writing this follow up question I'm not violating any rules. If I am I apologize.*

Let $G$ and $H$ be finite groups and $X$ and $Y$, $G$ and $H$-sets respectively. The main question is still the following:

Question:What kind of natural functors are there between categories of equivariant sheaves $Sh_G(X) \to Sh_H(Y)$?

Let's work with sheaves of sets and suppose for the moment $X$ and $Y$ points (Let's work with right actions throughout). In this case we're looking for functors: **$G$-Sets $\to H$-Sets**. The general yoga of **functors = bi-modules** gave me the idea to look inside the category of **$G^{op} \times H$-Sets**. Here's a nice step forward:

There's always a canonical functor

$$G \text{-Sets} \times (G^{op} \times H) \text{ -Sets} \to H \text{-Sets}$$

Which on objects acts by taking the "tensor product" $(A,P) \mapsto A \times_G P$ (note that we modded out by the right action of $G$ on $A$ and the left action on $P$ so now we only have a right $H$ action).

In the case of sheaves of vector spaces ($X$ and $Y$ still points) this is in fact precisely the familiar identification of functors with bimodules if we take the corresponding group algebras.

Suppose $X$ and $Y$ are arbitrary $G$ and $H$ sets now. I'd very much like to generalize the above to this situation but am still confused as to how to do so. Ideally I would like the following to hold and to have a concrete description as in the case where everything is a point:

Let $X$ and $Y$ as before. Every $G \times H$ equivariant sheaf on the product $X \times Y$ gives a pair of natural functors $Sh_G(X) \to Sh_H(Y)$ and $Sh_H(Y) \to Sh_G(X)$. And any interesting functors between these categories is in fact of this type.

Is there a way to make this true?