On the Kegel–Wielandt $\sigma$-Problem

For an arbitrary partition $\sigma$ of the set $\mathbb{P}$ of all primes, a sufficient condition for the $\sigma$-subnormality of a subgroup in a finite group is given. It is proved that, if a complete Hall set of type $\sigma$ is reduced into a subgroup $H$ of a $\sigma$-complete finite group $G$ all of whose non-Abelian composition factors are alternating groups, Suzuki groups, or Ree groups, then $H$ is $\sigma$-subnormal in $G$.