Formula of an injector of a finite $\pi$-soluble group

Let $G$ be a finite $\pi$-soluble group. We say that a Fitting set $\mathcal{F}$ of $G$ is $\pi$-saturated if it verifies $H\in\mathcal{F}$ whenever $O^{\pi'}(H)\in\mathcal{F}$. It is proved that $\mathcal{F}$-injector of $G$ is a subgroup of the form $W\cdot C_{D_p}(W/W_{F(p)})$, where $\mathcal{F}$ is a $\pi$-saturated Fitting set, which is defined with full integrated $H$-function $F$ of $G$, $\Sigma$ — Hall system of $G$, $D=N_G(\Sigma)$, $p\in\pi(G)\cap\pi\ne\varnothing$, $D_p\in\Sigma\cap D$, $W$ is an $\mathcal{F}$-injector of $O^p(G)$ and $\Sigma\searrow W$.