Strongly permutably complemented subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed strongly permutably complemented in $$G$$ if there exists a permutable complement $$K$$ to $$H$$ in $$G$$, such that, for any subgroup $$L \le H$$, $$K$$ and $$L$$ are permuting subgroups.

Stronger properties

 * Weaker than::Retract

Weaker properties

 * Stronger than::Permutably complemented subgroup
 * Stronger than::Lattice-complemented subgroup

Metaproperties
If $$H \le K \le G$$ are such that $$H$$ is strongly permutably complemented in $$K$$ and $$K$$ is strongly permutably complemented in $$G$$, then $$H$$ is strongly permutably complemented in $$G$$.