Let $G$ be a group and $H_1 ,H_2$ be two r-images of $G$; i.e. there exist two homomorphisms $f_i :H_i \to G$ and $g_i :G\to H_i$ such that $g_i f_i =id_{H_i}$, for $i=1,2$.
My question: Under which group conditions is the homomorphism $g_2 f_1 :H_1 \to G\to H_2$ an isomorphism?
Remember that $H_i$ is an r-image of $G$ iff $G={\rm im}f_i \ltimes {\rm ker}g_i$.
What I've tried: For example, I could prove that if ${\rm im}f_1={\rm im}f_2$ and ${\rm im}f_1$ is finite, then $g_2 f_1$ is an isomorphism. Also, I could show that if $G$ is indecomposable in terms of semidirect factors (I mean if $G=H\ltimes N$, then $N=1$ or $N=G$), then $f_1$ and $g_2$ are both isomorphism, which is really strong result because in this situation we have $G\cong H_i$.
I know that an epimorphims between two Hopfian isomorphic groups is an isomorphism but I don't know when $g_2 f_1$ is an epimorphism.
I have to say that I am working on the number of r-images of an arbitrary group up to isomorphism and my question comes from it.