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One restriction, the characteristic polynomials of finite-order elements in $GL(n,\mathbb{Z})$ should be products of cyclotomic polynomials. So an element of $GL(n,\mathbb{Z}/p)$ cannot be the image of a finite-order element of $GL(n,\mathbb{Z})$ if its characteristic polynomial is not a $\mod p$ reduction of a degree $n$ polynomial which is a product of cyclotomics.