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 of a finite group is given. It is proved that the Kegel–Wielandt $\sigma$-problem has a positive solution in the class of all finite groups all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank $1$.