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The problem of verifying the nonnegativity of a real valued function on a finite set is a long-standing challenging problem, which has received extensive attention from both mathematicians and computer scientists. Given a finite set X together with a function F : X → R, if we equip X a group structure G via a bijection ϕ : G → X, then effectively verifying the nonnegativity of F on X is equivalent to computing a sparse Fourier sum of squares (FSOS) certificate of f = F • ϕ on G. In this paper, we show that by performing the fast (inverse) Fourier transform and finding a local minimal Fourier support, we are able to compute a sparse FSOS certificate of f on G with complexity O(|G| log |G| + |G|t 4 + poly(t)), which is quasi-linear in the order of G and polynomial in the FSOS sparsity t of f . We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of order up to 10 6 . It is also noticeable that different choices of group structures on X would result in different FSOS sparsities of F . Along this line, we investigate upper bounds for FSOS sparsities with respect to different choices of group structures, which generalize and refine existing results in the literature. More precisely, (i) we give an upper bound for FSOS sparsities of nonnegative functions on the product and the quotient of two finite abelian groups respectively; (ii) we prove the equivalence between finding the group structure for the Fourier-sparsest representation of F and solving an integer linear programming problem.