Normal-potentially relatively characteristic subgroup

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Definition

A subgroup $H$ of a group $K$ is termed a normal-potentially relatively characteristic subgroup if there exists a group $G$ containing $K$ such that $K$ is a normal subgroup of $G$ , and any automorphism of $G$ that restricts to an automorphism of $K$ also restricts to an automorphism of $H$ .

Relation with other properties

Stronger properties

Weaker properties

Normal-extensible automorphism-invariant subgroup
Normal subgroup
Potentially relatively characteristic subgroup is a weaker notion, which turns out to be equivalent to normality. For full proof, refer: Potentially relatively characteristic equals normal