Variant of Hopf's formula for nilpotent multiplier of nilpotent group that uses the free nilpotent group of class equal to the sum

Statement
Suppose $$c_1,c_2$$ are (possibly equal, possibly distinct) natural numbers.

Suppose $$G$$ is a nilpotent group of nilpotency class $$c_1$$. We can calculate the $$c_2$$-nilpotent multiplier of $$G$$ as follows. Let $$F$$ be a free nilpotent group of class $$c_1 + c_2$$ and $$R$$ a normal subgroup of $$F$$ such that $$G \cong F/R$$. The Schur multiplier $$M^{c_2}(G)$$ is defined as:

$$M^{c_2}(G) \cong (R \cap (R \cap \gamma_{c_2+1}(F))/\gamma_{c_2+1}(R,F)$$.

Here:


 * $$\gamma_{c_2+1}(F)$$ is the $$(c + 1)^{th}$$ member of the lower central series of $$F$$, which is described explicitly as a $$(c+1)$$-fold iterated commutator of copie of $$F$$. Inductively, it is defined as $$\gamma_1(F) = F$$ and $$\gamma_{i+1}(F) = [F,\gamma_i(F)]$$
 * $$\gamma_{c_2+1}(R,F)$$ is defined as the $$(c_2+1)^{th}$$ member of the series defined inductively as $$\gamma_1(R,F) = R$$ and $$\gamma_{i+1}(R,F) = [F,\gamma_i(R,F)]$$

This is a variant of Hopf's formula for nilpotent multiplier. The original formula simply uses the free group. We replace it here by the group that is free of sufficiently large nilpotency class.