Characteristic Subgroup of Normal Subgroup is Normal

Theorem
Let $G$ be a group.

Let $N\leq G$ be normal.

Let $H\leq N$ be characteristic.

Then $H$ is normal in $G$.

Proof
Let $g\in G$.

Because $N$ is normal, conjugation by $g$ is an automorphism of $N$.

Because $H$ is characteristic in $N$, $gHg^{-1}=H$.

Thus $H$ is normal in $G$.