In 2014, Xu and Du classified all regular covers of a complete bipartite graph K n , n minus a matching, denoted by K n , n − n K 2 , whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of K n , n − n K 2 , whose covering transformation group is isomorphic to Z p 2 with p a prime and whose fibre-preserving automorphism group acts 2 -arc-transitively. Actually, there are only few covers with these properties and it is shown that all of them are covers of K 4, 4 − 4 K 2 .