Symmetric colorings of G × Z_2

Let G be a finite group and let r ∈ N . An r -coloring of G is any mapping χ : G → { 1 , . . . , r } . A coloring χ is symmetric if there is g ∈ G such that χ ( gx − 1 g ) = χ ( x ) for every x ∈ G . We show that if f ( r ) is the polynomial representing the number of symmetric r -colorings of G , then the number of symmetric r -colorings of G × Z 2 is f ( r 2 )