Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $$G,H,K$$ satisfying the following conditions:

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

• 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? May 14, 2016 at 7:27
• @user51223 Here is the reference jstor.org/stable/2042568?seq=1#page_scan_tab_contents May 16, 2016 at 18:14
• @IliasA. I have undeleted this question which you have deleted last year. -- Please do not self-delete your useful questions! Apr 30, 2021 at 20:55
• Do you have an example of an infinite group K and a finite subgroup G such that the inclusion induces an isomorphism in cohomology? May 11, 2021 at 12:18