Let $G$ be a finite (even cyclic group generated by $g$) and $X$ be a set on which it acts and considering the permutation representation induced by this, let $\chi_i$ be the set of characters. Then, the number of fixed points for instance is given by $|G|^{-1}(\sum_i \chi(g)$) and similarly, Burnside's lemma will give you a formula for the number of orbits.
My question is: is there an auxiliary representation we can construct such that the character of this rep evaluated at a generator (or a linear combination over evaluations at various elements) is one (or a constant that only depends on the group) exactly when the original representation has a fixed point and is 0 otherwise?