Finite groups with given local sections

A group is called primary if it is a finite $p$-group for some prime $p$. If $\sigma=\{\sigma_i\mid i\in I\}$ is some partition of $\mathbb{P}$, that is, $P=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$, then we say that a finite group $G$ is: $\sigma$-primary if it is a $\sigma_i$-group for some $i$; $\sigma$-nilpotent if $G=G_1\times\dots\times G_n$ for some $\sigma$-primary groups $G_1,\dots,G_n$. If $N=N_G(A)$ for some primary non-identity subgroup $A$ of $G$, then we say that $N/A_G$ is a local section of $G$. In this paper, we study a finite group $G$ under hypothesis that all proper local sections of $G$ belong to a saturated hereditary formation $\mathfrak{F}$, and we determine the normal structure of $G$ in the case when all local sections of $G$ are $\sigma$-nilpotent.