  We study the differential Galois theory of difference equations under weaker
hypothesis on the field of constants of the automorphism. This framework yields
a new approach to results by C.Hardouin and M.Singer, which answers possitively
a question by M.Singer: under the classical hypothesis, the known results are
still valid. In particular, our Galois group is isomorphic to theirs over a
suitable field. We also explicitly calculate the number of connected components
of the Galois group.
