The Brauer group of character rings

Let G be a finite group and R(G) be its character ring. Since R(G) is a finite Z-algebra, it is known that the Brauer group of the character ring is a finite sum of copies of Z/2Z. In this note, we determine the Brauer group of the character ring R(G), on showing how it is related to the conjugacy classes of G and the values of the irreducible characters on these classes.