  Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set
$\Sigma $ of subgroups of $G$ is called a \emph{$G$-covering subgroup system}
for the class ${\frak F}$ if $G\in {\frak F}$ whenever $\Sigma \subseteq {\frak
F}$.
  We prove that: {\sl If a set of subgroups $\Sigma$ of $G$ contains at least
one supplement to each maximal subgroup of every Sylow subgroup of $G$, then
$\Sigma$ is a $G$-covering subgroup system for the classes of all
$\sigma$-soluble and all $\sigma$-nilpotent groups, and for the class of all
$\sigma$-soluble $P\sigma T$-groups.}
  This result gives positive answers to questions 19.87 and 19.88 from the
Kourovka notebook.
