Abstract: A subgroup H of a finite group G is called a CAP^*-subgroup of G, if H either covers or avoids every non-Frattini chief factor of G. A subgroup H of G is said to be a nearly CAP^*-subgroup of G if there exists a subnormal subgroup K of G such that HK=G and H∩K is a CAP^*-subgroup of G. This paper investigates the structure of G under the assumption that certain subgroups of prime power orders are nearly CAP^*-subgroups of G, and a series of known results are generalized.

Key words: finite groups, nearly CAP^*-subgroup, p-supersolvable