In my research, I have to deal with the following cohomology:

Let $G$ be the semigroup of Hilbert polynomials, which as a subset of $\mathbb{Q}[x]$, consists of the Hilbert polynomials of all subschemes of $\mathbb{P}^N$ for various $N$, and the semigroup structure is given by product of polynomials. A theorem of Macaulay characterizes this semi-group completely. Consider $\mathbb{Z}/(2)$ as a trivial $G$-module. I have to calculate $H^2(G,\mathbb{Z}/(2))$.

I try to apply the universal coefficient theorem to do a reduction, but due to a lack of knowledge about $G$, I can't get further information.

Is it possible to calculate this group explicitly?

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