Normal-potentially relatively characteristic subgroup

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 $$H$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Characteristic-potentially characteristic subgroup
 * Weaker than::Normal-potentially characteristic subgroup

Weaker properties

 * Stronger than::Normal-extensible automorphism-invariant subgroup
 * Stronger than::Normal subgroup
 * Potentially relatively characteristic subgroup is a weaker notion, which turns out to be equivalent to normality.