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

Definition
A subgroup $$K$$ of a group $$G$$ is termed a normal subgroup in which every subgroup characteristic in the whole group is characteristic if $$K$$ is a normal subgroup of $$G$$ and every subgroup $$H$$ of $$K$$ that is a characteristic subgroup of $$G$$ is also a characteristic subgroup of $$K$$.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Normal intermediately AEP-subgroup
 * Weaker than::Normal AEP-subgroup

Weaker properties

 * Stronger than::Subgroup in which every subgroup characteristic in the whole group is characteristic
 * Stronger than::Normal subgroup