Cover-avoidance properties and the structure of finite groups

We call a subgroup A of a finite group G a CAP-subgroup of G if for any chief factor H/K of G, we have H∩A=K∩A or HA=KA. In this paper, some characterizations for a finite group to be solvable are obtained under the assumption that some of its maximal subgroups or 2-maximal subgroups be CAP-subgroups. We also determine the p-solvability and p-nilpotency of finite groups by considering their CAP-subgroups.