To add to Dbchatto67's answer: An overall idea of coming up with a counterexample is as follows:
Let $H_1$ be a cyclic group of non-prime odd order $q$ (and generator $\alpha$) so that it has a normal subgroup.
Let $H_2$ be a simple group with an element $\sigma$ of order $q$ i.e., the alternating group $A_q$.
Then $\phi: H_1 \mapsto H_2$ where $\phi(\alpha^i) = \sigma^i$ is well-defined, and furthermore, $\phi(H_1)$ is isomorphic to $H_1$; it is not a normal subgroup in the larger group $A_q$.
In general let $H_1$ be any group with a normal subgroup and let $H_2$ be any group that has as a subgroup $H'_1$ that satisfies (a) $H'_1$ isomorphic to $H_1$ and (b) $H'_1$ is not normal in $H_2$. Then the isomorphism from $H_1$ to $H'_1$ will work.