A generalisation of Schenkmans theorem

Let $G$ be a finite group and let $mathfrak{F}$ be a hereditary saturated formation. We denote by $mathbf{Z}_{mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $mathfrak{F}$-central in $G$, that is, [ (H/K) rtimes (G/mathbf{C}_{G}(H/K)) in mathfrak{F}. ]A subgroup $A leq G$ is said to be $mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$mathfrak{F}$-subnormal in $G$, if there is a subgroup chain [ A = A_0 leq A_1 leq ldots leq A_n = G ] such that either $A_{i-1} trianglelefteq A_{i}$ or $A_i / (A_{i-1})_{A_i} in mathfrak{F}$ for all $i = 1, ldots , n$. In this paper, we prove the following generalisation of Schenkmans Theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let $mathfrak{F}$ be a hereditary saturated formation and let $S$ be a $K$-$mathfrak{F}$-subnormal subgroup of $G$. If $mathbf{Z}_{mathfrak{F}}(E) = 1$ for every subgroup $E$ of $G$ such that $S leq E$ then $mathbf{C}_{G}(D) leq D$, where $D = S^{mathfrak{F}}$ is the $mathfrak{F}$-residual of $S$.