This is joint work with Ted Chinburg and Alex Lubotzky. Let F_d be a free discrete group of rank d > 1, and let \hat{F}_d be its profinite completion. Grunewald and Lubotzky developed a method to construct, under some technical conditions, representations of finite index subgroups of Aut(F_d) that have as images certain large arithmetic groups. In this talk, I will show how we can apply their method to Aut(\hat{F}_d). In this case, we obtain a stronger result in which we can describe much more precisely the images of the constructed representations and without any assumption. I will also discuss an application of this result to Galois theory. This uses a result by Belyi who showed that there is a natural embedding of the absolute Galois group G_Q of Q into Aut(\hat{F}_2). In particular, I will show how the natural action of certain subgroups of G_Q on the Tate modules of generalized Jacobians of covers of P^1 over \overline{Q} that are unramified outside {0,1,\infty} can be extended, up to a finite index subgroup, to an action of a finite index subgroup of Aut(\hat{F}_2).