Variant of Hopf's formula for Schur multiplier for powered nilpotent group that uses the free powered nilpotent group of class one more

Statement
Suppose $$\pi$$ is a set of primes and $$G$$ is a $$\pi$$-powered nilpotent group. Let $$F$$ be a free pi-powered nilpotent group of class $$c + 1$$ and $$R$$ be a normal subgroup of $$F$$ such that $$G \cong F/R$$. Then:

$$M(G) \cong (R \cap [F,F])/[F,R]$$

This is a variant of Hopf's formula for Schur multiplier of powered nilpotent group, and is similar to how variant of Hopf's formula for Schur multiplier for nilpotent group that uses the free nilpotent group of class one more is deduced from Hopf's formula for Schur multiplier.