  In this paper, we prove new instances of the inverse Galois problem over
global function fields for finite groups of Lie type. This is done by
constructing compatible systems of $\ell$-adic Galois representations valued in
a semisimple group $G$ using Galois theoretic and automorphic methods, and then
proving that the Galois images are maximal for a set of primes of positive
density using a classical result of Larsen on Galois images for compatible
sytems.
