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Given $S<\\text{AGL}(1,\\Bbb F_q)$ and an integer $k$ with $1\\le k\\le q$, does there exist a subset $B\\subset\\Bbb F_q$ with $|B|=k$ such that $S=\\text{AGL}(1,\\Bbb F_q)_B$? ($\\text{AGL}(1,\\Bbb F_q)_B=\\{\\sigma\\in\\text{AGL}(1,\\Bbb F_q):\\sigma(B)=B\\}$ is the stabilizer of $B$ in $\\text{AGL}(1,\\Bbb F_q)$.) We derive a sum that holds the answer to this question. 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