Given a finite group $G$ and a (finitely generated) $\mathbb{Z}G$-module $M$, assume that for each prime $p$ dividing the order $|G|$ of $G$ the $\mathbb{Z}\_pG$-module $M^{\mathbb{Z}\_p} = M\otimes\mathbb{Z}\_p$ is projective.



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Looking for criterion for $\mathbb{Z}G$-modules to be projective
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. 


