Injectors and Fischer subgroups of finite $\pi$-soluble groups

Let $\mathscr{F}$ be a Fitting set of a group $G$ and $L\leq G$. Then $\mathscr{F}$ is called a Fischer set of $G$, if $L\in\mathscr{F}$, $K\unlhd L$ and $H/K$ is a $p$-subgroup of $L/K$ for some prime $p$, then $H\in\mathscr{F}$. A subgroup $F$ of a group $G$ is said to be Fischer $\mathscr{F}$-subgroup of $G$ if the following conditions are hold: (1) $F\in\mathscr{F}$; (2) if $F\leq H\leq G$, then $H_{\mathscr{F}}\leq F$. Let $\pi$ be some nonempty set of prime numbers. A Fitting set $\mathscr{F}$ of a group $G$ is said to be $\pi$-saturated if $\mathscr{F}=\{H\leq G: H/H_{\mathscr{F}}\in\mathfrak{E}_{\pi'}\}$, where $\mathfrak{E}_{\pi'}$ is the class of all $\pi'$-groups. In this paper it is proved that if $\mathscr{F}$ is a $\pi$-saturated Fischer set of a $\pi$-soluble group $G$, then a subgroup $V$ of a group $G$ is $\mathscr{F}$-injector of $G$ if and only if $V$ is a Fischer $\mathscr{F}$-subgroup of $G$, which contains Hall $\pi'$-subgroup of $G$.