A subgroup $C$ of a group $G$ is said to be a characteristic subgroup of $G$ if and only if $T[C] \subset C$ for all automorphisms $T$ of $G$.
Here $T[C] \colon= \{ T(c) \colon c \in C \}$ is the image of set $C$ under the mapping $T$.
I can show that every characteristic subgroup of $G$ is a normal subgroup of $G$.
Does the converse hold too? If so, how to prove it? If not, what counter-example can be given?