Let $F_r$ be the free group on $r$ generators, let $G$ and $H$ be finite groups, and let $F_r\xrightarrow{\alpha}H\xleftarrow{\beta}G$ be surjective homomorphisms. It is then easy to see that we can choose $\phi\colon F_r\to G$ with $\beta\phi=\alpha$, but $\phi$ need not be surjective, especially if $G$ is too big. Now suppose in addition that there exists a surjective homomorphism $F_r\to G$. Is it then possible to find a surjective homomorphism $\phi\colon F_r\to G$ with $\beta\phi=\alpha$?

I think that this works if $G$ is nilpotent, because we can then reduce to the case of $p$-groups, and use the Frattini quotient to detect surjectivity. However, it would be tidier if I could prove it without any hypothesis. It also works when $r=1$, but even that case requires some modular arithmetic and so is not completely trivial. On the other hand, I think that in practice a randomly chosen $\phi$ will be surjective with high probability.

alsoworks when $r=1$": but then $G$ is abelian and hence nilpotent, so isn't it a particular case of the preceding assertion? $\endgroup$1more comment