Toroidal queens graphs over finite fields

For each positive integer n, the toroidal queens graph may be described as a graph with vertex set Zn × Zn where every vertex is adjacent to those vertices in the directions (1, 0), (0, 1), (1, 1), (1,−1) from it. We here extend this idea, examining graphs with vertex set F × F , where F is a finite field, and any four directions are used to define adjacency. The automorphism groups and isomorphism classes of such graphs are found.