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You can use the pre-$\lambda$ ring identity $$ \big(\sum_n [\mathrm{Sym}^n(X)]t^n\big)\big( \sum_k [\wedge^k(X)](-t)^k\big) = 1.$$ So it is enough to observe that the generating series for the exterior powers is a polynomial, for any finite $G$-set $X$. Here $\wedge^k(X)$ is the $G$-set of $k$-element subsets of $X$.