Intersection-transitively permutable subgroup

Definition
A subgroup of a group is termed intersection-transitively permutable if its intersection with any permutable subgroup of the group is again permutable in the whole group.

Note that permutability is not finite-intersection-closed.

Stronger properties

 * Weaker than::Right-transitively permutable subgroup
 * Weaker than::Hereditarily permutable subgroup
 * Weaker than::Hereditarily normal subgroup
 * Weaker than::Central subgroup