Suppose that $G$ is a finite group, $M$ is a right $G$-set and $N$ is a left $G$-set. Then we have a simplicial set $B(M,G,N)$ whose $n$-simplicies are $M \times G^n \times N$.

Now suppose that $H \subseteq G$ is a subgroup. There is a natural inclusion $B(*,H,*) \hookrightarrow B(*,G,G/H)$. Both of these spaces have fundamental group $H$ and no higher homotopy groups, and the inclusion induces an isomorphism on fundamental groups, so it is a weak equivalence.

Question:Is there a way to write down an explicit inverse weak equivalence $B(*,G,G/H) \to B(*,H,*)$? In practice, I would be happy with just the $B_{\leq 2}(*,G,G/H) \to B_{\leq 2}(*,H,*)$ part of such a map.

My instincts tell me that the answer is no: simplicial sets are not sufficiently squishy enough to do this, but I want to be sure before I start trying harder stuff.