Local Fitting sets and the injectors of a finite group

The product $\mathfrak{F}\diamond \mathfrak{X}$ of the Fitting set $\mathfrak{F}$ of a group $G$ and the Fitting class $\mathfrak{X}$ is called the set of subgroups $\{H\leq G:H/H_{\mathfrak{F}}\in \mathfrak{X}\}$. Let $P$ be the set of all primes, $\varnothing\neq \pi\subseteq P, \pi'=P\setminus \pi$ and $\mathfrak{G}_{\pi'}$ denote the class of all $\pi'$-groups. Let $\mathfrak{G}$ and $\mathfrak{G^{\pi}}$ to denote the class of all soluble groups and the class of all $\pi$-soluble groups, respectively. In the paper, it is proved that $\mathfrak{F}$-injector of a group $G$ either covers or avoids every chief factor of $G$ if $G$ is a partially soluble group. Chief factors of a group covered by $\mathfrak{F}$-injectors are described in the following cases: $1) G\in \mathfrak{F}\diamond \mathfrak{G}$; and $\mathfrak{F}$ is the Hartley set of $G$; $G\in \mathfrak{G^{\pi}}$ и $\mathfrak{F}=\mathfrak{F}\diamond \mathfrak{G_{\pi'}}$ for the integrated $H$-function $f$.