Centerless implies NSCFN in automorphism group

Statement

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

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

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