Consider the following problem:
Find integers $x_1, x_2, x_3,\dots, x_n$
Such that:
$$P(x_1,x_2,\dots, x_n) = Q$$
for some integer $Q$ and polynomial $P$ where for all permutations of any set of complex numbers $u_1, u_2, u_3,\dots,u_n$ $P(u_1, u_2, u_3,\dots,u_n)$ retains the same value (commutative with respect to all variables)
Can this be considered a variant of the Abelian Hidden Subgroup Problem?
If my understating of the definition of the Abelian Sub Group Problem is correct then I believe yes
Furthermore does that mean that this problem admits a polytime solution via Quantum Computer