Let $G$ be a group. A subgroup $H$ of $G$ is called characteristic if $\varphi(H)\subset H$ for all automorphisms $\varphi$ of $G$.
Now it is easy to show that every characteristic group is normal, as for every inner automorphism $\tau_g$, $\tau_g(H)\subset H$, i.e. $g^{-1}Hg\subset H$ for all $g\in G$.
Is that converse true? That is, whether every normal subgroup is characteristic?