Find existence of surjective homomorphism throug epimorphism onto each coset

Question

I have a little trouble with understanding cosets correctly.

I have a group $G$ (finitely generated) and a group $H$ finitely presented with the normal subgroup $Z \cong \mathbb{Z}$. It holds: $[H:Z] < \infty$.

Now, I know that there exists an epimorphism $\phi: G \to H/Z$. Therefore, I know that each coset $a\mathbb{Z} \in H/Z$ is hit through $\phi$. Does this mean that there exists an epimorphism $\theta: G \to H$?

Thanks a lot

Answer 1

No, it doesn't. Suppose that $G=\mathbb{Z}_2$, that $H=\mathbb{Z}$ and that $Z=2\mathbb{Z}$. Then there is an epimorphism from $G$ onto $H/Z$, namely $\phi(n)=n+2\mathbb{Z}$ ($n\in\{0,1\}$). But there is no eopimorphism from $G$ onto $H$ (since $G$ is finite and $H$ is infinite).