On some aspects of the Kegel-Wielandt $\sigma$-problem

For a partition $\sigma$ of the set $\mathbb{P}$ of all primes, a sufficient criterion for $\sigma$-subnormality of a subgroup in finite group is given. It is proved, that Kegel-Wielandt $\sigma$-problem has a positive solution in the class of all finite groups, in which all non-abelian composition factors are either alternating groups, or Suzuki groups, or sporadic groups.