There is a functorial reformulation of the theory of higher group characters.
For $k\in \mathbb{N}$, $k\ge 1$, let ${\Pi}_{1}^{k}({S}^{1})$ be the groupoid with $k$ objects which is weakly equivalent with respect to the folk model structure to the fundamental groupoid of the circle ${S}^{1}$.
In particular ${\Pi}_{1}^{1}({S}^{1})=B\mathbb{Z}$.
Then
Proposition. The $k$-conjugacy classes of $G$ are precisely the isomorphism classes of the groupoid
In particular the projection map
is precisely the corresponding weak Cayley table.
For $k=1$ we have that
is the “loop groupoid” of $G$ as discussed in
Simon Willerton , …,
i.e. the action groupoid of $G$ acting on itself by conjugation.