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.