An infinite series of regular edge- but not vertex-transitive graphs

Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q - 1, or n = 2 and q odd, we construct a connected q-regular edge-but not vertex-transitive graph of order 2qⁿ⁺¹. This graph is defined via a system of equations over the finite field of q elements. For n = 2 and q = 3, our graph is isomorphic to the Gray graph.