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

Given a nonempty set $\pi$ of primes, call a nilpotent group $\pi$-bounded whenever it has a central series whose every factor $F$ is such that: In every quotient group of $F$ all primary components of the torsion subgroup corresponding to the numbers in $\pi$ are finite. We establish that if $G$ is a residually $\pi$-bounded torsion-free nilpotent group, while a subgroup $H$ of $G$ has finite Hirsh–Zaitsev rank then $H$ is $\pi'$-isolated in $G$ if and only if $H$ is separable in $G$ in the class of all finite nilpotent $\pi$-groups. By way of example, we apply the results to study the root-class residuality of the free product of two groups with amalgamation.