Maximal subclasses of local fitting classes

A Fitting class $\mathfrak F$ is said to be $\pi$-maximal if $\mathfrak F$ is an inclusion maximal subclass of the Fitting class $\mathfrak S_\pi$ of all finite soluble $\pi$-groups. We prove that $\mathfrak F$ is a $\pi$-maximal Fitting class exactly when there is a prime $p\in\pi$ such that the index of the $\mathfrak F$-radical $G_\mathfrak F$ in $G$ is equal to 1 or $p$ for every $\pi$-subgroup of $G$. Hence, there exist maximal subclasses in a local Fitting class. This gives a negative answer to Skiba's conjecture that there are no maximal Fitting subclasses in a local Fitting class (see [1, Question 13.50]).