On partial CAP-subgroups of finite groups

Given a chief factor H/KH/K of a finite group G, we say that a subgroup A of G   avoids H/KH/K if H∩A=K∩AH∩A=K∩A; if HA=KAHA=KA, then we say that A   covers H/KH/K. If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p  -subgroup of order exceeding pkpk. If every subgroup of order pkpk, where k≥1k≥1, and every subgroup of order 4 (when pk=2pk=2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.