On sylowizers of some subgroups in finite groups

In this paper, we investigate the structure of a finite group [Formula: see text] by assuming that the intersections between [Formula: see text]-sylowizers of some [Formula: see text]-subgroups and a normal subgroup satisfy [Formula: see text]-permutability in [Formula: see text]. We obtain some criteria for [Formula: see text]-supersolvability and [Formula: see text]-nilpotency.