Geometric realisation of the graphs of McKay–Miller–Širáň

McKay, Miller and Širáň used a voltage graph construction to introduce three families of graphs of order 2q2, where q is a prime power. These graphs have diameter 2 and some of the largest known graphs of diameter 2 and given degree come from these families. We provide an alternative description of these graphs as modified incidence graph of an affine plane. This leads to a complete determination of their automorphism groups.