The automorphisms of certain Kummer surfaces act on the points mod $p$ by the alternating group

For the family of Kummer surfaces of the square of an elliptic curve over the prime field $\mathbb{F}_p$ with $p$ odd, we show that if the automorphism group acts transitively on the set of points then the action is at least alternating. This is analogous to a result of Meiri, Puder, and Chen arXiv:1702.08358 on the points mod $p$ of the Markoff surface.