Large orbits on Markoff-type K3 surfaces over finite fields

We study the surface ${\mathcal{W}}_k:x^2+y^2+z^2+x^2{y}^2{z}^2=kxyz$ in $({\mathbb{P}}^1{)}^3$, a tri-involutive K3 (TIK3) surface. We explain a phenomenon noticed by Fuchs, Litman, Silverman, and Tran: over a finite field of order $\equiv 1$ mod $8$, the points of ${\mathcal{W}}_4$ do not form a single large orbit under the group $\mathrm{\Gamma }$ generated by the three involutions fixing two variables and a few other obvious symmetries, but rather admit a partition into two $\mathrm{\Gamma }$-invariant subsets of roughly equal size. The phenomenon is traced to an explicit double cover of the surface.