Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 1 / a / ii / Solution

Put $H={\mathrm{Stab}}_F(x)$. For $g,h\in G$,Thus $f_x$ is constant on every left coset of $H$ and takes different values on different cosets. Its oracle is therefore an oracle for the hidden subgroup problem, and the hidden subgroup is precisely the stabilizer subgroup $H$.