We have a exact sequence of $G-$modules $$0\to I_G \to \mathbb{Z}G\to \mathbb{Z}\to 0$$ here $\varepsilon: \mathbb{Z}G \to \mathbb{Z}: \sigma\to 1 \;\forall \sigma \in G$ and $I_G=\ker\varepsilon$ or we can describe $I_G$ as free $\mathbb{Z}-$module with basis $\{\sigma-1: \sigma \in G\}$. Furthermore, this is a split $\mathbb{Z}-$sequence.
I want to construct a similar exact sequence of $G-$module $$0\to \mathbb{Z} \to \mathbb{Z}G \to J \to 0$$ How can I construct $\mathbb{Z} \to \mathbb{Z}G$ and what can $J$ be?
A $G$-module morphism $\mathbb{Z}\to \mathbb{Z}[G]$ is determined by the choice of a fixed point in $\mathbb{Z}[G]$ (for the $G$-action). These fixed points are exactly the elements $n\cdot \rho$ where $n\in \mathbb{Z}$ and $\rho=\sum_{\sigma\in G}\sigma$.
So you just have to choose some $n\in \mathbb{Z}$, define $\mathbb{Z}\to \mathbb{Z}[G]$ by $1\mapsto n\cdot \rho$, and then $J$ must be the quotient $\mathbb{Z}[G]/\langle n\cdot \rho\rangle$.