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.