Finite groups with weakly ℋ-subgroups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text] A subgroup [Formula: see text] of [Formula: see text] is said to be a weakly [Formula: see text]-subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text]. In this paper, we investigate the influence of weakly [Formula: see text]-subgroups on the structure of a finite group. Some recent results can be generalized.