Hilbert series of representations of categories of $G$-sets

Let $G$ be a finite group. A contravariant functor from the category of finite free $G$-sets to vector spaces has an associated Hilbert series, which records the underlying sequence of $G^n$ representations, $n\in \mathbb{N}$. We prove that this Hilbert series is rational with denominator given by linear polynomials with coefficients in the field generated by the character table of $G$.