Finite groups whose minimal subgroups are weakly H*-subgroups

Let G be a finite group. A subgroup H of G is called an H-subgroup in G if N_{G}(H)∩H^{g}≤H for all g∈G. A subgroup H of G is called a weakly H^{∗}-subgroup in G if there exists a subgroup K of G such that G=HK and H∩K is an H-subgroup in G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of G of prime order p or of order 4 (if p=2) is a weakly H^{∗}-subgroup in G. Our results improve and extend a series of recent results in the literature.