Counterexample: Two groups $H$ and $G$, with surjective homomorphismus

Question

So I need to find a counter-example, i.e. I need to find two groups $H$ and $G$, with $N$ being normal subgroup of $G$, with a NON-surjective Homomorphism $\phi: G \rightarrow H$, such that $\phi(N)$ is NOT normal subgroup of H.

I know for sure, that H isn´t supposed to be abelian group, but I am not sure which one to take! I would appreciate any kind of help.

Answer 1

Every group homomorphism $\Bbb Z\to S_3$ which maps $1$ into an odd permutation.