Suppose we have a finitely presented group $G$ with free presentation
$$
R\hookrightarrow F \twoheadrightarrow G.
$$
To this presentation we may associate an extension with abelian kernel
$$
R/R' \hookrightarrow F/R' \twoheadrightarrow G.
$$
Here $R'$ denotes the commutator subgroup of $R$, and the $G$-module $R/R'$ is sometimes known as the *relation module* associated to the presentation.

Under what conditions is the group $F/R'$ finitely presentable?

To give a concrete example, consider the free product of cyclic groups $C_2\ast C_3$ with presentation $\langle a,b \mid a^2,b^3 \rangle$.

Is there a nice finite presentation of $F/R'$ in this case?

I would also be interested in any pointers to places in the literature where calculations for particular presentations are carried out.