Matrix similar to a permutation matrix over $Z$

Question

Suppose we have $m\times m$ matrices $B,C$ with integer entries such that: $$A^{-1}BA=C.$$ for some matrix $A$ with integer entries. ($A^{-1}$ may not have integer entries). Now, suppose we also know that $C$ is a permutation matrix and that $det(A)$ is co-prime to the order of $C$ (as a permutation). Then, does there necessarily exist $A'\in GL_m(\mathbb{Z})$, so that: $$A'^{-1}BA'=C?$$ This question has its roots in the representation theory of finite groups. I know that the above isn't true if we remove the condition that $C$ is a permutation matrix. For a counterexample, see this. Does having this extra condition change the answer?