On the F-hypercentre and the intersection of all F-maximal subgroups of a finite group

We investigate the influence of the intersection of the F-maximal subgroups on the structure of a finite group. In particular, answering a question of L.A Shemetkov, we give conditions under which a hereditary saturated formation F has the property that for any finite group G, the F-hypercentre of G coincides with the intersection of all F-maximal subgroups of G.