Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: \mathrm{B}H \to \mathrm{B}G$ of (probably I should say "derived") stacks.

I would like to describe $\mathrm{B}G$ in terms of some sort of "groupoid" with base $\mathrm{B}H$. In some sense I know how to do this. If $f: N \to M$ were a surjective submersion of spaces, then $M$ would be equivalent to the "relative pair groupoid" of $N$, which you should try to build by taking the fibers of $f$ and taking the pair groupoid of each. I think you can write the nerve of this groupoid by saying that it has object space $M$, and space of 1-simplicies $M \times_N M$, and space of 2-morphisms $M \times_N M \times_N M$, and so one.

So I assume I can describe a stack equivalent to $\mathrm{B}G$ in terms of a simplicial object whose $k$-simplices are a $(k+1)$-fold homotopy-fibered-product, aka derived intersection, of $\mathrm{B}H$ with itself inside $\mathrm{B}G$.

But that's not very explicit, because I have no idea what derived intersections of classifying stacks of algebraic groups look like.

I'd much rather say something like "$\mathrm{B}G$ is the quotient groupoid for the action of SOMETHING on $\mathrm{B}H$". For comparison, if I have a surjection $H \to G$ of groups, then the underlying space $G$ is equivalent to the quotient groupoid of the space $H$ modulo the action of $\mathrm{ker}(H \to G)$ acting on $H$ by left (say) multiplication.

So my real hope is to do something with, say, the derived kernel of the inclusion $H \to G$. Not that I really know what those are in this nonabelian world. In my case, $H$ is not a normal subgroup of $G$, but nevertheless my hope-against-hope is to use some part of the cokernel of $H \to G$, which should have to do with the shift "$\mathrm{B}$".

How close are these hopes to reality?