I'm reading this paper, it says(in Theoerm 2) that:
Let $A$ be a finite abelian group, $G$ a finite group, $\Delta G$ the augmentation ideal of the integral group ring $\mathbb Z G$ and a $G$-modules extension $A\hookrightarrow B \twoheadrightarrow \Delta G$, then taking a free resolution of $B$ we get a commutative diagram (4) below:
How is $\mathbb Z G^{m}$ constructed that maps correspondingly?
$\mathbb{Z}G^m$ is the direct sum of $m$ copies of the group ring $\mathbb{Z}G$. The map $\mathbb{Z}G^m\twoheadrightarrow\Delta G$ is just mapping $1$ in each summand to corresponding generators of the ideal and extend by $\mathbb{Z}G$-linearity.