This work presents a difference geometric approach to the strongly normal Galois theory of difference equations. In this approach, a system of ordinary difference equations is encoded in a difference extension, and the Galois groups are group schemes of finite type over the constants. The Galois groups need neither be linear nor reduced. The main result is a characterization of the extensions that admit a reasonable Galois theory by a normality property. This approach has been inspired by the recent work of J. Kovacic on the differential Galois theory of strongly normal extensions.