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.