I am searching for either (1) a reference of what's currently known or (2) a general outline to the approach to answering the following question. (I will also accept special cases, such as the case $d=1$, though I don't know if that really helps.)
Let $\mathbb{F}_q$ denote the finite field of $q=p^d$ elements. The root question is: what is the size of the conjugacy class of a given matrix $A \in \mathrm{GL}_n(\mathbb{F}_q)$?
Besides writing out the generic form of a conjugate of $A$ and trying to count them, I've made attempts by switching to centralizers by the orbit-stabilizer theorem. If $C(A)$ is the centralizing subgroup for $A$, we have that the size of the conjugacy class of $A$ is the index $[\mathrm{GL}_n(\mathbb{F}_q): C(A)]$. Since the order of $\mathrm{GL}_n(\mathbb{F}_q)$ is known, this gives my answer if only I can compute the order of $C(A)$. This is where I'm stuck. I've also tried thinking about canonical forms, but encounter field problems. It seems like working out who commutes with $A$ should be something known, but maybe this is deceptively difficult. Or, as in my usual style, I have missed something trivial or obvious.
Any references to the literature or paths to approach this question are appreciated.