On the Kegel-Wielandt σ-problem for binary partitions

Let sigma = {sigma(i) : i is an element of I} be a partition of the set P of all prime numbers. A subgroup X of a finite group G is called sigma-subnormal in G if there is a chain of subgroups X = X-0 subset of X-1 subset of ... subset of X-n = G where for every j = 1, ..., n the subgroup Xj-1 is normal in X-j or X-j/Core(Xj)(Xj-1) is a sigma(i)-group for some i is an element of I. In the special case that sigma is the partition of P into sets containing exactly one prime each, the sigma-subnormality reduces to the familiar case of subnormality. A finite group G is sigma-complete if G possesses at least one Hall sigma(i)-subgroup for every i is an element of I, and a subgroup H of G is said to be sigma(i)-subnormal in G if H boolean AND S is a Hall sigma(i)-subgroup of H for any Hall sigma(i)-subgroup S of G. Skiba proposes in the Kourovka Notebook the following problem (Question 19.86), that is called the Kegel-Wielandt sigma-problem: Is it true that a subgroup H of a sigma-complete group G is sigma-subnormal in G if H is sigma(i)-subnormal in G for all i is an element of I? The main goal of this paper is to solve the Kegel-Wielandt sigma-problem for binary partitions.