The Frattini subgroup is a characteristic subgroup.

HINT. If $H$ is a maximal subgroup of $G$, and $f\colon G\to K$ is a homomorphism with kernel contained in $H$, then $f(H)$ is a maximal subgroup of $f(G)$.

let $\alpha$ be an arbitrary automorphism of $G$. If $M$ be a maximal subgroup of $G$, then $\alpha(M)$ is a maximal subgroup, too. then $\alpha(\phi(G))=\alpha(\cap M)=\cap \alpha(M)=\cap M'=\phi(G)$, M' is a maximal subgroup.