Injectors in Fitting Sets of Finite Groups

A set of subgroups $\mathscr F$ of a finite group $G$ is referred to as a Fitting set if it is closed with respect to taking normal subgroups, products of normal $\mathscr F$-subgroups, and inner automorphisms of $G$. A Fitting set $\mathscr F$ of a group $G$ is said to be $\pi$-saturated if $H\in\mathscr F$ for every subgroup $H$ in $G$ such that $O^{\pi'}(H)\in\mathscr F$. In the paper, it is proved that, if $\mathscr F$ is a $\pi$-saturated Fitting set of a $\pi$-solvable group $G$, then there are $\mathscr F$-injectors in $G$ and every two of them are conjugate.