Separability of subgroups of nilpotent groups in the class of finite π-groups

Let π be a nonempty set of primes. We prove that a nilpotent group possesses the property of separability of all its π′-isolated subgroups in the class of finite π-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 π are finite. We prove that the converse holds too for torsion-free nilpotent groups.