Characteristically simple and normal fully normalized implies minimal normal

Statement

If $H$ is a Normal fully normalized subgroup (?) of $G$ and $H$ is characteristically simple as a group, then $H$ is a Minimal normal subgroup (?) of $G$.

Facts used

1. Normal upper-hook fully normalized implies characteristic: If $K\leq H\leq G$ are such that $K$ is normal in $G$ and $H$ is fully normalized in $G$, then $K$ is characteristic in $H$.

Proof

Given: A group $G$, a characteristically simple, normal and fully normalized subgroup $H$.

To prove: $H$ is a minimal normal subgroup of $G$.

Proof: By assumption $H$ is normal, so it suffices to show that any normal subgroup $K$ of $G$ contained in $H$ is either trivial or equal to $H$. Let's do this.

By fact (1), $K$ is characteristic in $H$. Since $H$ is characteristically simple, we see that $K$ must be either equal to $H$, or trivial, completing the proof.