$H$ is a characteristic subgroup of $K$ if $\Phi(H)=H~\forall~\Phi\in Aut(K).$ Prove that if $H$ is a characteristic subgroup of $K$, and $K$ is a normal subgroup of $G$, then $H$ is a normal subgroup of $G$

**Attempt:** Suppose $h \in H, k \in K$

Given that $\Phi(H) = H ~~\forall~~ \Phi~\in Aut(K).$

Since, $Inn (K) \subset Aut(K) \implies k^{-1}Hk = H \implies H \vartriangleleft K$ .... $(1)$

Also, it's given that $K \vartriangleleft G$ .....$(2)$

From $(1),(2): H \vartriangleleft K \vartriangleleft G$

But, I don't think this means that $H \vartriangleleft G?$ Is there something which I am missing?

Thank you for your help