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

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