Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, analogous to induction for an inclusion.)
"Natural" I mean ... maybe left or right adjoint to the pullback from $Rep(G/N)$ to $Rep(G)$. I don't know exactly.
If not, then maybe there is a good reason why such a thing cannot exist?
I'd like to ask the the more general question: if $G$ and $E$ are finite, connected groupoids, $G \to E$ a functor, and $F : G \to Vect$ a functor. Can $F$ be pushed forward to a functor on $E$?
As motivation: I would like to understand how to build the induced bundle construction categorically: we start with a representation of $H$, a functor $F$ from $BH$ (groupoid of $H$ torsors) to $Vect$, and try to push $F$ forward along the inclusion of $BH$ into the action groupoid $G/H$. (Then to get the induction construction, "pushforward" again to $BG$.)
The question A construction of a pushforward of Vect-presheves. popped up while writing this question. There is a link to a paper of Jeffrey Morton which seems to indicate the answer to the general question is something like yes, if I'm understanding correctly (which almost surely I am not). It seems that the construction I am looking for is that of a Kan extension, at least in the case of an inclusion of groups. I'm not really sure what is going on.
There are two natural things you can do, corresponding to the left and right adjoint of restriction. Explicitly, the left adjoint sends a $G$-representation $V$ to the $G/N$-representation
$$V/(nv - v, n \in N)$$
and the right adjoint sends a $G$-representation to the $G/N$-representation
$$\{ v \in V : nv = v \forall n \in N \}.$$
In other words, the left adjoint takes $N$-coinvariants and the right adjoint takes $N$-invariants. Both of these are equipped with a $G/N$-action, and the left / right adjointness describes a universal property of this $G/N$-action that you should write down if you haven't already.
More generally, there are left and right adjoints to pullback along a functor between two groupoids, not necessarily finite or connected, or for that matter between two categories. They're called left and right Kan extension, and loosely speaking they correspond to taking fiberwise colimits and fiberwise limits respectively.
It's a good exercise to write down what this looks like for the restriction of scalars functor $\text{Mod}(S) \to \text{Mod}(R)$ coming from a map $f : R \to S$ of rings (e.g. group rings). The left adjoint is extension of scalars / induction and the right adjoint is a different thing that you might call coextension of scalars or coinduction.