Let $G$ be a group. A subgroup $H$ of $G$ is called characteristic if $\phi(H ) ⊂ H$ for all automorphisms $ϕ$ of $G$. Pick out the true statement(s):
(a) Every characteristic subgroup is normal.
(b) Every normal subgroup is characteristic.
(c) If $N$ is a normal subgroup of a group $G$, and $M$ is a characteristic sub-group of $N$, then $M$ is a normal subgroup of $G$.
I am completely stuck on it. Can somebody help me to solve the problem?