Schur multiplier preserves powering for nilpotent groups

Statement
Suppose $$\pi$$ is a set of primes and $$G$$ is a $$\pi$$-powered nilpotent group. Then, the Schur multiplier $$M(G)$$ is also $$\pi$$-powered.

Related facts

 * Exterior square preserves divisibility for nilpotent groups
 * Schur multiplier of divisible nilpotent group need not be divisible by any prime

Facts used

 * 1) uses::Homology and localization commute