Nonorientable regular embeddings of graphs of order p2

In [J. Algeb. Combin. 19(2004), 123–141], Du et al. classified the orientable regular embeddings of connected simple graphs of order pq for any two primes p and q. In this paper, we shall classify the nonorientable regular embeddings of these graphs, where p 6= q. Our classification depends on the classification of primitive permutation groups of degree p and degree pq but is independent of the classification of the arc-transitive graphs of order pq.