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It seems to me that the question is equivalent to classifying up to conjugacy in ${\rm GL}(n,\mathbb{Q}),$ those matrices $M$ which satisfy $M^{d} = \frac{u}{v}I,$ where $d$ is a positive integer, $u$ is a positive integer, $v$ is an integer, ${\rm gcd}(u,v) = 1$ and neither $u$ nor $v$ is divisible by the $d$-th power of any prime, and furthermore, no lower power of $M$ is scalar. Then it becomes a question of determining what the possible rational canonical forms for $M$ can be.