Hopf's formula for Schur multiplier of powered nilpotent group

Statement
Let $$\pi$$ be a set of primes.

Let $$G$$ be a group isomorphic to the quotient group $$F/R$$, where $$F$$ is a free pi-powered group and $$G$$ is a $$\pi$$-powered nilpotent group. Note that this implies that $$R$$ is also a $$\pi$$-powered group.

Then, the Schur multiplier of $$G$$ is given as:

$$M(G) = H_2(G;\mathbb{Z}) \cong (R \cap [F,F])/[R,F]$$

In other words, it equals the quotient of the intersection of $$R$$ with the commutator subgroup of $$F$$ by the focal subgroup of $$R$$ in $$F$$. ($$[R,F]$$ equals the focal subgroup because $$R$$ is a normal subgroup of $$F$$).

Note that any choice of generating set (in the $$\pi$$-powered sense) for $$G$$ gives a choice of $$F$$ and $$R$$ for which the theorem can be applied: $$F$$ is the free group on those generators with the natural surjection, and $$R$$ is the kernel of the surjection.

Related facts

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