# Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions

1. $G$ and $H$ are finite groups and $K$ an infinite group.
2. 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.

• Does the proof of the case $K$ being finite has anything to do with some Eilenberg-Moore type spectral sequence? And how a comparison map $G\to H$ is constructed in the finite case? Is it constructed geometrically? – user51223 May 14 '16 at 7:27
• @user51223 Here is the reference jstor.org/stable/2042568?seq=1#page_scan_tab_contents – Ilias A. May 16 '16 at 18:14