This is a partial answer. Frobenius investigated such Fourier coefficients to prove that the number of solutions to $x^d=1$ is divides the order of the group. He showed your class function is a virtual character, so all your Fourier coefficients are integers. The paper of Frobenius is in German so I can't read it. The paper http://u.cs.biu.ac.il/~vishne/publications/S0219498811004690.pdf, and its references, should give you some of what you want.

This is a partial answer. Frobenius investigated such Fourier coefficients to prove that the number of solutions to $x^d=1$ is divides the order of the group. He showed your class function is a virtual character, if I understood correctly, so all your Fourier coefficients are integers. The paper of Frobenius is in German so I can't read it. The paper http://u.cs.biu.ac.il/~vishne/publications/S0219498811004690.pdf, and its references, should give you some of what you want.