Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions
- $G$ and $H$ are finite groups and $K$ an infinite group.
- there exists two monomorphisms $G\rightarrow K\leftarrow H$ inducing an isomorphism in homology (with integral coefficients)
My question is the follwoing: It there a know triple $(K, G, H)$ verifiying the conditions 1 and 2 such that $G$ is not isomorphic to $H$ ? I'm more interested when $G$ is a perfect group, but any example (if there exists) is welcome.
Edit: In the case where $K$ is finite, Culler theorem says that $K, G, H$ are all isomorphic.