Finite groups with some primary subgroups g-s-supplemented

Let F be a class of groups. A subgroup H of a group G is called F-s-supplemented in G if there exists a subgroup K of G such that G = HK and K/K boolean AND H-G belongs to F. We obtain some results about the F-s-supplemented subgroups and use them to determine the structure of some groups. In particular, some new criteria of p-nilpotency, solubility, supersolubility of a group are obtained.