Centerless implies NSCFN in automorphism group

Statement

Suppose $G$ is a Centerless group (?). Then, consider the homomorphism $G\to \mathrm{A}\mathrm{u}\mathrm{t}(G)$, given by the action by conjugation. Then, the homomorphism is injective. Identifying $G$ with its image, $\mathrm{I}\mathrm{n}\mathrm{n}(G)$, we obtain that $G$ is a subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(G)$. This subgroup is a NSCFN-subgroup (?), i.e.:

1. $G$ is a normal subgroup inside $\mathrm{A}\mathrm{u}\mathrm{t}(G)$.
2. $G$ is a self-centralizing subgroup inside $\mathrm{A}\mathrm{u}\mathrm{t}(G)$, i.e., $C_{\mathrm{A}\mathrm{u}\mathrm{t}(G)}(G)\le G$ (in fact, $G$ is a centralizer-free subgroup).
3. $G$ is a fully normalized subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(G)$: Every automorphism of $G$ extends to an inner automorphism of $\mathrm{A}\mathrm{u}\mathrm{t}(G)$.

Related facts

• Every group is normal fully normalized in its holomorph
• Abelian implies self-centralizing in holomorph
• Additive group of a field implies monolith in holomorph
• Characteristically simple and non-abelian implies monolith in automorphism group
• Characteristically simple and non-abelian implies automorphism group is complete