Universal property of quotient group to get epimorphism I know there are already answers to the question of the universal property of quotient groups. For example here: Universal property of quotient group.
My question is now: If I have the homomorphism: $\phi: G \to G'$ and let H be $H\subset ker(\phi)$. Moreover I have the canonical projection $\pi: G \to G/H$. Due to the univeraly property of quotient groups there exists a unique homomorphism: $\bar{\phi}: G/H \to G'$.
Can I now say that: I know that $\pi$ is surjective. The homormophism $\phi$ is surjective iff $\bar{\phi}$ is surjective?

Yes. In fact, this has nothing to do with group theory. If you have a surjective map $f\colon A\longrightarrow B$ and you have a map $g\colon B\longrightarrow C$, then $g$ is surjective if and only if $g\circ f$ is surjective.
