The representation theory of finite groups over the fields $\mathbb{C}$ or $\mathbb{Q}$ is clear. I am wondering what happens if we consider something similar over the ring $\mathbb{Z}$. To be more precise, let $G$ be a finite subgroup of $GL_n(\mathbb{Z})$, and we consider the action of $G$ on $\mathbb{Z}^n$.
$\textbf{Question 1}$: How to check that there are no nontrivial $G$-invariant subgroups of $\mathbb{Z}^n$?
$\textbf{Question 2}$: Is it possible to describe $G$-invariant subgroups of $\mathbb{Z}^n$?
Could you suggest a good reference dealing with such questions?