Analogue of orbit counting formula in $F[x]$

Question

We know that if $G$ is a finite group acting on a finite set $\Omega$, then $$\sum_{g\in G}\chi(g)\equiv 0\pmod{|G|}$$ where $\chi$ is the permutation character. I wonder if is there an analogue of this formula in $F[x]$ where $F$ is a finite field. More clearly, given a finite group $G$, do we have a formula with a polynomial $f$ such that $$\sum_{g\in G}\chi(g)\equiv 0\pmod f$$ where $\chi$ is some sort of a polynomial valued function.