A characteristic subgroup is a normal subgroup

let $\phi$ be inner automorphism for example $\phi_g(x)=g^{-1}xg$ for any $g\in G$. Now you can apply this automorphism.

$\phi_g(H)=H$ $\Longrightarrow$ for any $h\in H$ then we have $\phi_g(h)\in H$. Hence $H^g=H$ for any $g\in G$. So $H\trianglelefteq G$.

$\phi (H)=H$ for all automorphism
We wanted to show that H is normal subgroup that is for every $x \in G$ $xHx^{-1}=H$ Now as we have $\phi_x(g)=xgx^{-1}$ as inner automorphism .$\phi_x(H)=xHx^{-1}=H$ Hence Done