Normal fully normalized subgroup

Symbol-free definition
A subgroup of a group is termed normal fully normalized if it is normal in the whole group, and every automorphism of the subgroup extends to an inner automorphism of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed normal fully normalized if $$H$$ is normal in $$G$$, and every $$\sigma \in \operatorname{Aut}(H)$$ can be realized as conjugation by $$g$$ for some $$g \in G$$.

Equivalently, $$H$$ is normal in $$G$$, and the natural map $$G \to \operatorname{Aut}(H)$$ is surjective.

Conjunction with other properties

 * Weaker than::NSCFN-subgroup is the conjunction with the property of being a self-centralizing subgroup.

Weaker properties

 * Stronger than::Normal subgroup
 * Stronger than::Fully normalized subgroup

Facts

 * Every group is normal fully normalized in its holomorph
 * Normal upper-hook fully normalized implies characteristic
 * Characteristically simple and normal fully normalized implies minimal normal

Left realization
Every group can be realized as a normal fully normalized subgroup in some group, for instance, in its holomorph.