It is well-known that for a finite group $G$ and field $k$ of characteristic 0, the linearization morphism $B(G) \to R\_k(G)$ has in most cases nontrivial kernel, and this can be used to find permutation modules which admit non-isomorphic permutation bases (as $G$-sets). I'm hoping to find a relatively easy-to-state example, but this time, for a permutation $\mathbb{Z} G$-module instead.


Unfortunately, all the examples I've cooked up over $\mathbb{Q}$ haven't extended to $\mathbb{Z}$. Given that Krull-Schmidt doesn't hold for $\mathbb{Z}G$ in general, I'm almost certain that examples of such permutation modules exist - does anyone have any examples of this phenomenon?



Examples of permutation $\mathbb{Z}G$-modules which admit non-isomorphic permutation bases?
A similar question was asked on [math.stackexchange](https://math.stackexchange.com/q/2730330) a few years ago, and I posted the following answer. I've just looked again at Conlon's paper, and I'm afraid it's a bit short on explicit examples.


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There are fairly general examples due to Conlon in the paper:


*Conlon, S.B.*, [**Monomial representations under integral similarity**](http://dx.doi.org/10.1016/0021-8693(69)90111-2), J. Algebra 13, 496-508 (1969). [ZBL0185.06702](https://zbmath.org/?q=an:0185.06702).


There is even a transitive example, due to Scott:


*Scott, Leonard L.*, Integral equivalence of permutation representations, Sehgal, Surinder (ed.) et al., Group theory. Proceedings of the 21st biennial Ohio State-Denison mathematical conference, Granville, OH (USA), 14-16 May, 1992. Singapore: World Scientific. 262-274 (1993). [ZBL0828.20004](https://zbmath.org/?q=an:0828.20004).


In this example, $G$ is $\text{PSL}(2,29)$ and the permutation actions are on the cosets of two non-conjugate subgroups both isomorphic to the alternating group $A\_5$.


