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.

Edit: We can assume that $H$ has decidable membership problem. In the context that I am working in, $H$ will be the centralizer of a previous matrix so membership is solvable by the above logic. I also have no idea if anything even exists in the case $H = GL(n, \mathbb{Z})$.