Picard-Vessiot extensions with specified Galois group

Picard-Vessiot extensions are determined by their differential module structure. For a fixed group G, Picard-Vessiot extensions with differential Galois group G are all isomorphic as G-modules but not as differential rings. We show that isomorphism classes of Picard-Vessiot extensions with group G correspond to G-orbits in a certain finite-dimensional vector space with G-action.