  In previous work, the first author established a natural bijection between
minimal subshifts and maximal regular J-classes of free profinite semigroups.
In this paper, the Sch\"utzenberger groups of such J-classes are investigated,
in particular in respect to a conjecture proposed by the first author
concerning their profinite presentation. The conjecture is established for all
non-periodic minimal subshifts associated with substitutions. It entails that
it is decidable whether a finite group is a quotient of such a profinite group.
As a further application, the Sch\"utzenberger group of the J-class
corresponding to the Prouhet-Thue-Morse subshift is shown to admit a somewhat
simpler presentation, from which it follows that it has rank three, and that it
is non-free relatively to any pseudovariety of groups.
