What is explicit form of this kernel?

Question

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by $(n,x) \mapsto x$ induces a map from $$\frac{(R \rtimes R) \cap ([S \rtimes F, S \rtimes F])}{[R \rtimes R, S \rtimes F]} \to \frac{R \cap [F, F]}{[R,F]}.$$ What is the kernel of this homomophism in explicit form?