Exterior square preserves divisibility for nilpotent groups

Statement
Suppose $$G$$ is a nilpotent group and $$\pi$$ is a set of prime numbers such that $$G$$ is $$\pi$$-divisible (i.e., $$G$$ is a $$\pi$$-divisible nilpotent group). The exterior square of $$G$$, denoted $$G \wedge G$$, is also a $$\pi$$-divisible nilpotent group.

Facts used

 * 1) uses::Stem extension preserves divisibility for nilpotent groups
 * 2) uses::Derived subgroup is divisibility-closed in nilpotent group

Proof using the Schur covering group
Recall that the exterior square $$G \wedge G$$ can be defined as the derived subgroup of any Schur covering group of $$G$$. The Schur covering group is itself nilpotent (of class at most one more than the class of $$G$$) and is a stem extension of $$G$$. Facts (1) and (2) give the result.