Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two representations are equivalent?
I know that if we were working over $\mathbf{C}$ then we could consider characters since these two representations have the same character and are therefore equivalent, but does this method survive the transition to integral representation?