On $\mathfrak{F}$-radicals of finite $\pi$-soluble groups

In this paper, we prove that for every local $\pi$-saturated Fitting class $\mathcal{F}$ with $char(\mathcal{F})=\mathbb{P}$, the $\mathcal{F}$-radical of every finite $\pi$-soluble groups $G$ has the property: $C_G(G_\mathcal{F})\subseteq G_\mathcal{F}$. From this, some well known results are followed and some new results are obtained.