I know this is exactly the same as this question. But the proof detailed uses restrictions, and I'm not familiar with that. The way I want to prove this is by the standard method of showing two sets are equal since both $\phi(G)$ and $G$ are both sets correct?
If we have groups $N,K,G$ where $N \leq K \leq G$.
$\phi(N) = N$ $\forall \phi \in Aut(K)$
$\phi(K) = K$ $\forall \phi \in Aut(G)$
$\phi(N) = N$ $\forall \phi \in Aut(G)$
First we pick $x \in \phi(N)$ and show $x \in N$ where $\phi: G \rightarrow G$ is an automorphism.
However, this is where I'm not sure what to do. Am I even on the right track here?