How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left multiplication?

Note: Suppose $S$ is freely generated by $\xi_1, \xi_2, \cdots, \xi_s$ then the clearly the problem is equivalent to describing the unimodular matrices fixing each of the tuples $\xi_1, \cdots \xi_s$. If the $\xi_j$s have a simple form, e.g., $\xi_j$s are the standard basis vectors $e_1, \cdots, e_n$ of $\mathbb Z^n$ then the form of matrices fixing each $\xi_j \ (j = 1, \cdots, s)$ is easy to determine. But in general the method to obtain the stabilizer is not clear.