I wish to prove whether this is true or false.

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> If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$.
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I'm not even sure if $N$ being normal in $G$ implies that $\phi(N)$ is normal in $H$, but I can't think of an immediately obvious counter example.