Let $M \in H \leq GL(n, \mathbb{Z})$. Is there an algorithm that computes either matrix generators or even a group presentation for $C_H(M)$ given generators or a presentation of $H$? Also is $C_H(M)$ necessarily finitely generated if $H$ is?

I realize that you can view $C_H(M)$ as the set of integer solutions of the equations $MAM^{-1} = A$ where $A \in H$, but I wasn't sure how one can find generators.