On dihedral flows in embedded graphs

Let $$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $$ , let $$ be the dihedral group of order $$ . Define $$ . Goodall et al in 2016 asked whether $$ admits a nowhere-identity $$ -flow if and only if it admits a nowhere-identity $$ -flow with $$ (a "nowhere-identity dihedral $$ -flow"). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere-identity $$ -flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true are described. We focus particularly on cubic graphs.