Products of F* (G)-subnormal subgroups of finite groups

A subgroup H of a group G is called F* (G)-subnormal if it is subnormal in the product with the generalized Fitting subgroup F* (G) of G. All saturated formations F that contain every group G = AB where A and B are F* (G)-subnormal F-subgroups are described. Moreover, if such formation F is also normally hereditary, then every group G = AB where A and B are F* (G)-subnormal quasi-F-subgroups is a quasi-F-group. The connection of above-mentioned formations to the lattice formations, CWP-formations and formations with the Kegel property is found.