On $\pi$-solvability of a finite group with a partially permutable $\pi$-Hall subgroup

We investigate a finite group such that its $\pi$-Hall subgroup is permutable with some subgroups of the group. For instance, we establish $\pi$-solvability of such groups in two cases: a $\pi$-Hall subgroup is 2-nilpotent; a $\pi$-Hall subgroup is solvable and $3\notin\pi$. Besides, we establish solvability of a finite group in the case when a 2-nilpotent $\pi$-Hall subgroup of the group has odd index.