In a couple of recent works with Khare and Larsen we have developed a method to construct finite Lie groups of type $C_n, B_n$ and $G_2$ (and potentially more) as Galois groups over the field of rational numbers. The main tools are functorial lifts and self-dual automorphic forms on $GL(n)$ as a source of $l$-adic representations (Kottwitz, Clozel, Harris-Taylor). In this talk I will discuss how one can control the image of the $l$-adic representation by picking local components of automorphic representations: deeply embedded tame parameters and Jordan subgroups, as main tools.