  Let $G$ be a finite group of Lie type defined in characteristic $p$, and let
$k$ be an algebraically closed field of characteristic $r>0$. We will assume
that $r \neq p$ (so, we are in the non-defining characteristic case). Let $V$
be a finite-dimensional irreducible left $kG$-module. In 2011, Guralnick and
Tiep found bounds on the dimension of $H^1(G,V)$ in non-defining
characteristic, which are independent of $V$. The aim of this paper is to
generalize the work of Gurlanick and Tiep. We assume that $G$ is split and use
methods of modular Harish-Chandra theory to find bounds on the dimension of
$\mathrm{Ext^1}$ between certain irreducible $kG$-modules. We then use Dipper
and Du's algorithms to illustrate our bounds in a series of examples.
