Finite groups with some maximal subgroups of Sylow subgroups ℳ-supplemented

A subgroup H of a group G is said to be ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let ℱ be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ℱ. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F*(H), having no supersolvable supplement in G, is ℳ-supplemented in G. Then G ε ℱ.