Let $G$ be a finite group, and $\rho_1, \rho_2: G\to GL_n(\mathbb C)$ be two representations. Suppose that $\rho_1$ and $\rho_2$ are equivalent (i.e. conjugate over $\mathbb C$), and suppose that both groups $\rho_1(G)$, $ \rho_2(G)$ belong to $GL(n,\mathbb Z)$. Is it true that these two groups are conjugate in $GL(n,\mathbb Z)$?
If not, is this at least true in the case when $G$ is a symmetric group $S_n$ and the representation $\rho$ is irreducible? The motivation for this question is the following: I know that all complex irreducible representations of $S_n$ can be defined over integers. I wonder whether there is somehow a canonical choice.