Finite groups with absolutely $\mathfrak{F}$-subnormal maximal subgroups

A subgroup $M$ of a group $G$ is an $n$-maximal subgroups of $G$ if there is a subgroup chain $M=M_n\leq M_{n-1}\leq \ldots \leq M_1\leq M_0=G$ such that $M_{i+1}$ is a maximal subgroup of $M_i$. We establish a criterion for a group with absolutely $\mathfrak{F}$-subnormal $n$-maximal subgroups to belong to a subgroup-closed saturated formation $\mathfrak{F}$ containing all nilpotent groups.