The functor of complex characters of finite groups

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.