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Here is an infinite collection of "cheating" counterexamples for $p=2$. Let $G$ be a cyclic group of order $2^n ,$ generated by $g$, say. Let $M$ be the augmentation ideal of the group ring $\mathbb{Z}_{2}G,$ so that $M = \sum_{x \in G} \alpha_x x,$ where $\sum_{x \in G}\alpha_{x} = 0$. Regard $M$ as a (say, right) $\mathbb{Z}_{2}G$-module (of rank $2^{n}-1$, for example with $\mathbb{Z}_2$-basis $\{ x - 1_G: 1 \neq x \in G \}$). Note that the minimum polynomial of $g$ on $M$ is $\frac{t^{2^n}-1}{t-1}$, and note also that $g$ acts with determinant $-1$ on $M.$ Now let $V$ be a rank 1 $\mathbb{Z}_2G$-module on which $g$ acts as $-1$. Then $M \otimes V$ and $M$ have isomorphic reductions (mod 2), but are not isomorphic as $\mathbb{Z}_2G$-modules (since $g$ acts with determinant $1$ on the first, and determinant $-1$ on the second).