Relatively characteristic subgroup

Definition with symbols
Let $$H \le K \le G$$ be groups. Then we say that $$H$$ is relatively characteristic in $$K$$ with respect to $$G$$ if any automorphism of $$G$$ that restricts to an automorphism of $$K$$ must in fact restrict to an automorphism of $$H$$.

Stronger properties

 * Weaker than::Middle-characteristic subgroup: If $$H \le K \le G$$ and $$H$$ is characteristic in $$K$$, it is relatively characteristic in $$K$$ with respect to $$G$$.
 * Weaker than::Characteristic subgroup: If $$H \le K \le G$$ and $$H$$ is characteristic in $$G$$, it is relatively characteristic in $$K$$ with respect to $$G$$.

Weaker properties

 * Stronger than::Normalizer-relatively normal subgroup
 * Stronger than::Relatively normal subgroup

A related subgroup property
A subgroup $$H$$ of a group $$K$$ is said to be potentially relatively characteristic in $$K$$ if there exists a group $$G$$ containing $$K$$ such that $$H$$ is relatively characteristic in $$K$$, relative to $$G$$. Potentially relatively characteristic subgroups are precisely the same as normal subgroups.