Have been in a course on group theory, and was hinted that there can be a group action by $D_4$ on the set of $2$-colorings of a hexagon.
But, that implies a homomorphism between the two groups $D_4, D_6.$
(This is shown by failure , by me to have a group action in the second table, due to the overlap of two orbits : $\mathcal O_6, \mathcal O_7$)
It is justified, in the Edit, by the failure of homomorphism between the two groups: $D_4, D_6$)
Hence, does it imply that the question can be changed to:
Under what conditions is it possible to define homomorphism between the two dihedral groups: $D_4, D_6?$
Kindly give hints, by providing under what conditions is it possible to have the needed group action.
Or, is it that have construed the hint wrongly?