How can we compute $|\mathrm{Aut}(M)|$ from a basis for $\mathrm{End}(M)$?

Question

For a finite field $\mathbf{k}$ with $|\mathbf{k}|=q$, and an object $M$ of a $\mathbf{k}$-linear category, suppose we have finite basis for $\mathrm{End}(M)$. How can we compute the size of $\mathrm{Aut}(M)$? What do we need to know about those basis elements to compute this?