Right-transitively permutable subgroup

Symbol-free definition
A subgroup of a group is termed right-transitively permutable if any permutable subgroup of the subgroup is also permutable in the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed right-transitively permutable if whenever $$K$$ is a permutable subgroup of $$H$$, then $$K$$ is permutable in $$G$$.

Stronger properties

 * Hall direct factor: A Hall subgroup that is also a direct factor

Weaker properties

 * Permutable subgroup