Finite groups with generalized subnormal embedding of Sylow subgroups

Given a set $\pi$ of primes and a hereditary saturated formation $\mathfrak F$, we study the properties of the class of groups $G$ for which the identity subgroup and all Sylow $p$-subgroups are $\mathfrak F$-subnormal ($\mathrm K$-$\mathfrak F$-subnormal) in $G$ for each $p$ in $\pi$. 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.