Orders of Cayley nut graphs with prescribed edge or arc orbit counts

Determine, for each integer k ≥ 2, all graph orders n for which (a) there exists a Cayley nut graph of order n with exactly k edge orbits and (b) there exists a Cayley nut graph of order n with exactly k arc orbits.

Background

The paper proves that Cayley nut graphs exist with any prescribed number k ≥ 2 of edge orbits and, separately, any prescribed number k ≥ 2 of arc orbits, and that infinitely many examples exist in every such case. However, the authors do not determine which individual orders support these configurations. Examples in the conclusion show that the known constructions do not provide a complete order characterization, motivating the explicitly posed realizability problem.