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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.

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  • $\begingroup$ 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? $\endgroup$ – user51223 May 14 '16 at 7:27
  • $\begingroup$ @user51223 Here is the reference jstor.org/stable/2042568?seq=1#page_scan_tab_contents $\endgroup$ – Ilias A. May 16 '16 at 18:14

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