Abelian implies self-centralizing in holomorph

Statement
Suppose $$G$$ is an abelian group and $$H = G \rtimes \operatorname{Aut}(G)$$ is its holomorph. Then, $$C_H(G) = G$$. In other words, $$G$$ is a fact about::self-centralizing subgroup of $$H$$.

Further, since every group is normal fully normalized in its holomorph, this tells us that $$G$$ is normal, self-centralizing and fully normalized in its holomorph. In short, $$G$$ is a fact about::NSCFN-subgroup of its holomorph, and thus, is a fact about::NSCFN-realizable group.

Applications

 * Additive group of a field implies monolith in holomorph

Other related facts

 * Center is a direct factor implies NSCFN-realizable
 * Characteristically simple implies NSCFN-realizable
 * Centerless implies NSCFN-realizable
 * Characteristically simple implies monolith in automorphism group