Counting symmetric colorings of G x Z2

Let G be a finite group and let r is an element of N. An r-coloring of G is any mapping chi : G -> {1, . . . ,r}. A coloring chi is symmetric if there is g is an element of G such that chi(gx(-1)g) = chi(x) for every x is an element of G. We show that if G is Abelian and f(r) is the polynomial representing the number of symmetric r-colorings of G, then the number of symmetric r-colorings of G x Z(2) is f(r(2)). We also extend this result to the dihedral group D(G).