Every group is normal fully normalized in its holomorph

Statement

Suppose $G$ is a group. Denote by $\operatorname {Hol} (G)$ the holomorph of $G$; in other words, we have:

$\operatorname {Hol} (G)=G\rtimes \operatorname {Aut} (G)$

with the natural action of the automorphism group $\operatorname {Aut} (G)$ on $G$. Then, $G$ is a normal fully normalized subgroup of $\operatorname {Hol} (G)$:

• $G$ is normal in $\operatorname {Hol} (G)$: Every inner automorphism of $\operatorname {Hol} (G)$ restricts to an automorphism of $G$.
• $G$ is fully normalized in $\operatorname {Hol} (G)$: Every automorphism of $G$ extends to an inner automorphism in $\operatorname {Hol} (G)$.