When does group homomorphism imply quasi-isometric

Question

I could not find a good source for this. Any help would be appreciated.

Definition:

Answer 1

A group homomorphism $f \colon G \to H$ between two finitely generated groups $G$ and $H$ is a quasi-isometry if and only if $ker(f)$ is finite and $im(f)$ has finite index in $H$. The proof is probably a good exercise (but not too hard after all).

This gives a characterization depending on the morphism $f$ in question.

The characterization you wrote does not involve the morphism $f$. Take e.g. two infinite finitely generated virtually isomorphic groups as you wrote and let $f$ be the trivial morphism. Then the groups are clearly quasi-isometric but $f$ is certainly not a quasi-isometry.