I'm looking for a reference that tells me either the centraliser of $\mathrm{O}^\epsilon(n,q)$ in $\mathrm{GL}(n,q)$, or how to describe the module homomorphisms of $\mathrm{O}^\epsilon(n,q)$ acting on its natural module.
I've tried some googling with no success, and all I've been able to infer so far is that $Z(\mathrm{GL}(n,q))$ is in the centralizer.
Any help would be much appreciated!