Finite groups with some CAP-subgroups

Let G be a finite group. A subgroup H of G is called a CAP-subgroup if the following condition is satisfied: for each chief factor K/L of G either HK = HL or H ∩ K = H ∩ L. Let p be a prime factor of |G| and let P be a Sylow p-subgroup of G. If d is the minimum number of generators of P then there exists a family of maximal subgroups of P, denoted by M _d (P)={P ₁, P ₂,…, P _d } such that ∩ _i=1 ^d P _i = ?(P). In this paper, we investigate the group G satisfying the condition: every member of a fixed M _d (P) is a CAP-subgroup of G. For example, if, in addition, G is p-solvable, then G is p-supersolvable.