Are there two non-isomorphic groups $G_1$ and $G_2$ so that $G_1$ is an r-image of $G_2$ and $G_2$ is an r-image of $G_1$?
Recall that a group $H$ is called an r-image of a group $G$ if there are homomorphisms $f$ and $g$ from $H$ to $G$ and from $G$ to $H$, respectively, so that $gf=id_H$.
I know that there are two non-isomorphic groups which are isomorphic to a subgroup of each other, like two free groups of rank 2 and 3.
Thanks in advance.
According to answers to this Math Overflow question, there is an Abelian group $A$ such that $A\cong A^3$ but $A\not\cong A^2.$ Clearly $A$ is an $r$-image of $A^2$ while $A^2$ is an $r$-image of $A^3\cong A$. Therefore, $A$ and $A^2$ are non-isomorphic groups which are $r$-images of each other.