Super-symmetric maps from generalized dihedral groups

G.A. Jones proposed a universal method of construction of super-symmetric maps (i.e., regular maps invariant under all hole  operators and dualities) with a given automorphism group $G$ by means of the intersection of kernels of all epimorphisms from a suitable universal group $U$ onto $G$. An equivalent way of describing this construction uses parallel products of maps, studied by S. Wilson. We explore this construction in the case when $G$ is a generalized dihedral group extended by a reflection. We show that in most cases there are 6 equivalence classes of epimorphisms $U \to G$, and hence $6$ different kernels, but in all these cases a parallel product of only $3$ maps with automorphism group $G$ is sufficient to generate a super-symmetric map.