  A Weierstrass Prym eigenform is an Abelian differential with a single zero on
a Riemann surface possessing some special kinds of symmetries. Such surfaces
come equipped with an involution, known as a Prym involution. They were
originally discovered by McMullen and only arise in genus 2, 3 and 4. Moreover,
they are classified by two invariants: discriminant and spin.
  We study how the fixed points for the Prym involution of Weierstrass Prym
eigenforms are permuted. In previous work, the authors computed the permutation
group induced by affine transformations in the case of genus 2, showing that
they are dihedral groups depending only on the residue class modulo 8 of the
discriminant $D$. In this work, we complete this classification by settling the
case of genus 3, showing that the permutation group induced by the affine group
on the set of its three (regular) fixed points is isomorphic to
$\mathrm{Sym}_2$ when $D$ is even and a quadratic residue modulo 16, and to
$\mathrm{Sym}_3$ otherwise. The case of genus 4 is trivial as the Pyrm
involution fixes a single (regular) point. In both cases, these same groups
arise when considering only parabolic elements of the affine group.
  By recent work of Freedman, when the Teichm\"uller curve induced by
Weierstrass Prym eigenform is not arithmetic, the fixed points of the Prym
involution coincide with the periodic points of the surface. Hence, in this
case, our result also classifies how periodic points are permuted.
