  We determine the structure of the fibered biset functor sending a finite
group $G$ to the complex vector space of complex valued class functions of $G$.
Previously, it is studied as a biset functor by Bouc and as a $\mathbb
C^\times$-fibered biset functor by Boltje and the second author. In this paper,
we complete the study by considering the other choices of the fiber group. As a
corollary, we obtain a (previously known) set of composition factors of the
biset functor of $p$-permutation modules.
