  In previous papers, we defined $q$-analogues of alien derivations for linear
analytic $q$-difference equations with integral slopes and proved a density
theorem (in the Galois group) and a freeness theorem. In this paper, we
completely describe the wild fundamental group and apply this result to the
inverse problem in $q$-difference Galois theory. The new version contains an
appendix on pronilpotent completion and the main result on the direct problem
is made more precise. (Submitted for publication)
