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Let $H$ be a hidden subgroup of $G_1$ that is indistinguishable from subgroup $H^{\prime}$ by quantum Fourier sampling. Now take a larger group $G_2$ such that it contains $G_1$. Now if I do quantum Fourier sampling over $G_2$ instead of $G_1$ are they still indistinguishable

Can something similar be said about hidden shift either? (This question was already asked on quantun stackexchange but did not recieve any answer so it was suggested to repost it on theory)

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