New 2-closed groups that are not automorphism groups of digraphs

In this paper we extend the construction of Giudici, Morgan and Zhou [arXiv:2110.07896] to give the first known examples of nonregular, $2$-closed permutation groups of rank greater than $4$ that are not the automorphism group of any digraph. We also show that this construction only gives examples for four particular primes.