See Curtis–Reiner's textbook on the Representation Theory of Finite Groups and Associative Algebras ^{(MR 144979)}, Theorem 74.3, page 507, and especially the introduction starting on page 493.

The result for cyclic groups of prime order, and for order 4 was originally done in:

- Diederichsen, Fritz-Erdmann.
"Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz"
Abh. Math. Sem. Hansischen Univ. 13, (1940). 357–412. MR2133.

However, Reiner has written quite a few nice papers on similar subjects. One of his earlier ones is on the same topic:

- Reiner, Irving. "Integral representations of cyclic groups of prime order."
Proc. Amer. Math. Soc. 8 (1957), 142–146.
MR83493
doi:10.2307/2032829

One can also consult texts on things called "crystallographic groups", "space groups", and "p-adic space groups". Plesken has written several nice books using this sort of thing. These give infinite families of nicely related finite groups and of course help crystallographers.

Be careful to distinguish these sorts of representations from ZG-modules. ZG-modules are basically incomprehensible, so instead lots of people focus on ZG-lattices, where the underlying module is projective. This means the idea of using matrices still makes some sense. There is a lot of literature on modules over group rings over nice rings (like Z or Dedekind domains), but a fair amount of it is not applicable to questions about GL(n,Z).

Roughly speaking, even for G=1, ZG modules are too difficult to understand, and adding a G just makes it worse. Another common tack is to look at $\hat {\mathbb{Z}}_p$ modules, p-adic modules. Again the results are nicest for lattices, but things do not get so bad near so fast there. Reiner's Maximal Orders textbook describes some of the beautiful and well-behaved things you can see there.