Kummer theory for Drinfeld modules

Let phi be a Drinfeld A-module of characteristic p(0) over a finitely generated field K. Previous articles determined the image of the absolute Galois group of K up to commensurability in its action on all prime-to-p(0) torsion points of phi, or equivalently, on the prime-to-p(0) adelic Tate module of phi. In this article we consider in addition a finitely generated torsion free A-submodule M of K for the action of A through phi. We determine the image of the absolute Galois group of K up to commensurability in its action on the prime-to-p(0) division hull of M, or equivalently, on the extended prime-to-p(0) adelic Tate module associated to phi and M.