Large orbits on Markoff-type K3 surfaces over finite fields

We study the surface $\mathcal{W}_k : x^2 + y^2 + z^2 + x^2 y^2 z^2 = k x y z$ 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 $\Gamma$ generated by the three involutions fixing two variables and a few other obvious symmetries, but rather admit a partition into two $\Gamma$-invariant subsets of roughly equal size. The phenomenon is traced to an explicit double cover of the surface.