As Geoff mentioned, there is a lot of material on this type of question in "Methods of Representation Theory" by Curtis and Reiner, though I think Volume 1 is more relevant here.

If we specialise Corollary (25.16) to the case you are interested in, then we get the following:

Let $G$ be a finite group of order $n$. Let $M$ be a finitely generated left $\mathbb{Z}[G]$-module. Suppose that $M$ is free as a $\mathbb{Z}$-module. Then $M$ is $\mathbb{Z}[G]$-projective if and only if $M \otimes_{\mathbb{Z}}\mathbb{Z}_{p}$ is $\mathbb{Z}_{p}[G]$-projective if each prime $p$ dividing $n$.

"Maximal Orders" by Reiner will probably also be a useful reference.

As Geoff mentioned, there is a lot of material on this type of question in "Methods of Representation Theory" by Curtis and Reiner, though I think Volume 1 is more relevant here.

If we specialise Corollary (25.16) to the case you are interested in, then we get the following:

Let $G$ be a finite group of order $n$. Let $M$ be a finitely generated left $\mathbb{Z}[G]$-module. Suppose that $M$ is free as a $\mathbb{Z}$-module. Then $M$ is $\mathbb{Z}[G]$-projective if and only if $M \otimes_{\mathbb{Z}}\mathbb{Z}_{p}$ is $\mathbb{Z}_{p}[G]$-projective for each prime $p$ dividing $n$.

"Maximal Orders" by Reiner will probably also be a useful reference.