Centerless implies NSCFN in automorphism group

Statement
Suppose $$G$$ is a fact about::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 fact about::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., $$C_{\operatorname{Aut}(G)}(G) \le G$$ (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