$\require{AMScd}$Let $G, H, K$ be groups and suppose that we have a diagram $$\begin{CD} G @>f_1>> H\\ @Vg_1VV\\ K \end{CD} $$ where morphisms $f_1$ and $g_1$ induce isomorphism in homology. Can one builds a group $U$ with the properties
- The following square commutes: $$\begin{CD} G @>f_1>> H\\ @Vg_1VV @VVg_2V\\ K @>>f_2> U \end{CD} $$
- $g_2$ and $f_2$ induce isomorphism in homology.
I tried to prove that $U:=K*_G H$ will be such a group. I used Mayer-Vietoris sequence, but it doesn't seem to be working.