Finite groups in which all $2$-maximal subgroups are $\pi$-decomposable

Let $\pi$ is a set of prime numbers. A very broad generalization of notion of nilpotent group is the notion of $\pi$-decomposable group, i.e. the direct product of $\pi$-group and $\pi'$-group. In the paper, the description of the finite non-$\pi$-decomposable groups in which all $2$-maximal subgroups are $\pi$-decomposable is obtained. The proof used the author's results connected with the notion of control the prime spectrum of finite simple groups. The finite nonnilpotent groups in which all $2$-maximal subgroups are nilpotent was studied by Z. Janko in 1962 in case of nonsolvable groups and the author in 1968 in case of solvable groups.