Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:
- $G$ and $H$ are finite groups and $K$ is an infinite group.
- there exist two monomorphisms $G \rightarrow K \leftarrow H$ which induce an isomorphism in homology (with integral coefficients).
My question is the following: Is there a known triple $(K, G, H)$ satisfying the conditions (1.) and (2.) such that $G$ is not isomorphic to $H$? I am more interested in the case that $G$ is a perfect group, but any example (if there exists such) is welcome.
Edit: In the case where $K$ is finite, Culler's theorem says that $K, G, H$ are all isomorphic.