$H$ is a subgroup of $G$ with $\phi(H) = H$ $\forall \phi \in Aut(G) \Rightarrow $ $H$ is normal in $G$ 
$H$ is a subgroup of $G$ with $\phi(H) = H$ $\forall \phi \in Aut(G) \Rightarrow $ $H$ is normal in $G$ 

I'm not sure where to begin. 
Perhaps we can use this corollary of the First Isomorphism Theorem 
$N_G(H)/C_G(H) \cong Aut(H)$ 
But I'm not sure if this is helpful.

Hint
$$h\mapsto ghg^{-1}$$
is an automorphism for all $g\in G$.
