Classification and properties of the $pi$-submaximal subgroups in minimal nonsolvable groups

Let $pi$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $pi$-submaximal subgroup if there is a monomorphism $phi:Xrightarrow Y$ into a finite group $Y$ such that $X^phi$ is subnormal in $Y$ and $H^phi=Kcap X^phi$ for a $pi$-maximal subgroup $K$ of $Y$. In his talk at the well-known conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the $pi$-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set $pi$ of primes, we obtain a description of the $pi$-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandts problem.