Subgroup in which every subgroup characteristic in the whole group is characteristic

Definition
Suppose $$K \le G$$ is a subgroup. We say that $$K$$ is a subgroup in which every subgroup characteristic in the whole group is characteristic if, for any subgroup $$H \le K$$ such that $$H$$ is a defining ingredient::characteristic subgroup of $$G$$, $$H$$ is also a characteristic subgroup of $$K$$.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::AEP-subgroup
 * Weaker than::Normal subgroup in which every subgroup characteristic in the whole group is characteristic
 * Weaker than::Characteristic subgroup in which every subgroup characteristic in the whole group is characteristic
 * Weaker than::Subgroup whose intersection with a characteristic subgroup of the whole group is characteristic in it