Separability of subgroups of nilpotent groups in the class of finite $\pi$-groups

Let $\pi$ be a nonempty set of primes. We prove that a nilpotent group possesses the property of separability of all its $\pi'$-isolated subgroups in the class of finite $\pi$-groups if it has a central series whose every factor $F$ satisfies the condition: In every quotient group of $F$, all primary components of the torsion subgroup corresponding to the numbers of $\pi$ are finite. We prove that the converse holds too for torsion-free nilpotent groups.