Sorry for a presumably noobish group theory question. I would like an example of the following:
$G_1,G_2$ are finitely presented groups such that there exists finitely presented groups $C_1,C_2$ with the property that $G_1 \twoheadrightarrow C_1 \leftarrowtail G_2$ and $G_2 \twoheadrightarrow C_2 \leftarrowtail G_1$ but $G_1 \ncong G_2$.
Maybe someone who knows a little group theory can put $F_3$ in a 2-generated finitely presented group? Or maybe that never happens...
I would also like to know of conditions on $G_1, G_2$ that imply that this can not happen. One condition is if either of the groups is abelian, but I would be interested in hearing other conditions.