Base sizes for finite groups of Lie type in cross-characteristic

Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we determine the minimal base size of $G$ when $E(G)/Z(E(G))$ is a finite simple group of Lie type in cross-characteristic. We show that $G$ has a regular orbit on $V$, with specific exceptions, for which we find the base size.