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 $$\operatorname{Hol}(G)$$.