Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity).

To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is essentially obvious that all standard operations on sheaves (pull,push,shriek push,tensor etc.) elevate to the level of $G$-equivariant sheaves on $G$-sets and with $G$-equivariant maps.

So as long as we remain in a context where everything has an action of $G$ things look very similar (almost identical) to the non-equivariant setting.

There are however other natural operations one can do with equivariant sheaves.

For instance let $X$ be a $G$-set and $p: X \to Y=X/G$ be the quotient map. In this case we have a natural functor which assigns to an equivariant sheaf $\mathcal{F}$ the $G$-invariant sections in the pushforward:

$$ \mathcal{F} \mapsto (p_*\mathcal{F})^G \in Sh(Y)$$

Alternatively one could replace invariants with coinvariants.

For a different example consider a $G$ set $X$ and a subset $S \in X$. Suppose $Stab_G(S)=H$. There's a natural functor $Sh_G(X) \to Sh_H(S)$ which is the usual pullback at the level of sheaves but remembers the $H$-equivariant structure. I think there should be a natural functor:

$$Sh_H(S) \to Sh_G(X)$$

But at this point i'm confused as to how to define it. In the particular case where $X$ is an orbit and $S=\{x\}$ is a point these functors should give a natural equivalence.

In a general I'd like to understand when and how do equivariant categories for different groups interact. Slightly more specifically: let $X$ be a $G$-set and $Y$ an $H$-set. And suppose $G$ and $H$ have some morphism between them (either $G \to H$ or $H \to G$) And suppose we have a map of sets $X \to Y$ (or the other way round) which respects this morphism.

What kind of natural functors are there between the categories $Sh_G(X)$ and $Sh_H(Y)$?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity).

To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is essentially obvious that all standard operations on sheaves (pull,push,shriek push,tensor etc.) elevate to the level of $G$-equivariant sheaves on $G$-sets and with $G$-equivariant maps.

So as long as we remain in a context where everything has an action of $G$ things look very similar (almost identical) to the non-equivariant setting.

There are however other natural operations one can do with equivariant sheaves.

For instance let $X$ be a $G$-set and $p: X \to Y=X/G$ be the quotient map. In this case we have a natural functor which assigns to an equivariant sheaf $\mathcal{F}$ the $G$-invariant sections in the pushforward:

$$ \mathcal{F} \mapsto (p_*\mathcal{F})^G \in Sh(Y)$$

Alternatively one could replace invariants with coinvariants.

For a different example consider a $G$ set $X$ and a subset $S \in X$. Suppose $Stab_G(S)=H$. There's a natural functor $Sh_G(X) \to Sh_H(S)$ which is the usual pullback at the level of sheaves but remembers the $H$-equivariant structure. I think there should be a natural functor:

$$Sh_H(S) \to Sh_G(X)$$

But at this point i'm confused as to how to define it. In the particular case where $X$ is an orbit and $S=\{x\}$ is a point these functors should give a natural equivalence.

In a general I'd like to understand when and how do equivariant categories for different groups interact. Slightly more specifically: let $X$ be a $G$-set and $Y$ an $H$-set. And suppose $G$ and $H$ have some morphism between them (either $G \to H$ or $H \to G$) And suppose we have a map of sets $X \to Y$ (or the other way round) which respects this morphism.

What kind of natural functors are there between the categories $Sh_G(X)$ and $Sh_H(Y)$?

EDIT: I phrased everything in the simple context of sets for clarity. I will prefer a natural answer that would generalize easily to any relevant context. Hopefully with some intuitive/geometric/hands-on interpretation of the the functors involved beyond formal existence arguments.