  We show that at any standard or Weierstrass point P on a hyperelliptic
Riemann surface S equipped with a nonsingular even spin structure, we may
attach a spin group G(P) in a natural way. All such spin groups are isomorphic
to each other and to the alternating group A(g)<S(g). Moreover, for any two
vertices of the same spin graph we construct a natural, unique isomorphism
between the corresponding, mutually conjugate spin groups.
