QUESTION:
Suppose $B$ is a symmetry group of a non-square rectangle $R$ that has vertices $W, X, Y, Z$ as well as edges $h = WX, p = XY, k = YZ$ and $n = ZW$. And let $A$ be the set of all $2$-colorings of the set of edges $\{h, p, k, n\}$ of $R$ on which $B$ acts in the usual manner. Find $|B|$ and $|A|$ and then find the number of distinct $B$-orbits on $A$ with Burnside’s Counting Theorem.
I'm struggling with this question, but gave it a go anyways.
I noted that $|B|$ has order $4$ as it belongs to Dihedral Group $2$. Same with $|A|$. Would I also be right to assume the stabiliser for $W, X, Y$ and $Z$ is the identity?
Finally I put down the orbit of $W$ to be $(W,X,Y,Z).$
That's all I've done so far. However, something feels off. Could someone let me know if I'm on the right track so far (particularly with my order on $A$, as I've never come across the term $2$-colorings before)?
Thanks