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

Statement
Suppose $$G$$ is a nilpotent group of nilpotency class $$c$$. We can calculate the Schur multiplier of $$G$$ as follows. Let $$F$$ be a free nilpotent group of class $$c + 1$$ and $$R$$ a normal subgroup of $$F$$ such that $$G \cong F/R$$. The Schur multiplier $$M(G)$$ can be computed as:

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

This is a variant of Hopf's formula for Schur multiplier. The original version of the formula stipulates that $$F$$ must be a free group. Note that, when generalizing, what's crucial is to have a class of one more than the class of $$G$$, because we need a quotient that is big enough to be sensitive to $$[F,R]$$.

Corresponding formula for exterior square
Suppose $$G$$ is a nilpotent group of nilpotency class $$c$$. We can calculate the Schur multiplier of $$G$$ as follows. Let $$F$$ be a free nilpotent group of class $$c + 1$$ and $$R$$ a normal subgroup of $$F$$ such that $$G \cong F/R$$. The exterior square $$G \wedge G$$ can be computed as:

$$G \wedge G \cong [F,F]/[F,R]$$