Finite groups with generalized subnormal embedding of Sylow subgroups

Given a set π of primes and a hereditary saturated formation F, we study the properties of the class of groups G for which the identity subgroup and all Sylow p-subgroups are F-subnormal (K-F-subnormal) in G for each p in π. We show that such a class is a hereditary saturated formation and find its maximal inner local screen. Some criteria are obtained for the membership of a group in a hereditary saturated formation in terms of its formation subnormal Sylow subgroups.