Hereditarily permutable subgroup

Symbol-free definition
A subgroup of a group is termed hereditarily permutable if every subgroup of that subgroup is a permutable subgroup in the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed hereditarily permutable in $$G$$ if whenever $$K \le H$$, $$K$$ is a permutable subgroup of $$G$$.

Stronger properties

 * Weaker than::Hereditarily normal subgroup
 * Weaker than::Central subgroup

Weaker properties

 * Stronger than::Right-transitively permutable subgroup
 * Stronger than::Intersection-transitively permutable subgroup
 * Stronger than::Permutable subgroup

Facts

 * The Baer norm of a group, defined as the intersection of normalizers of all its subgroups, is hereditarily permutable.