Subgroup for which every lattice complement is a permutable complement

Definition
A subgroup of a group is termed a subgroup for which every lattice complement is a permutable complement if it has the property that any lattice complement to it is a permutable complement to it.

In particular, if such a subgroup is a lattice-complemented subgroup (i.e., it has at least one lattice complement) then it is a permutably complemented subgroup.