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A group isomorphism $H$ defined between $\\mathcal A$ and $\\mathcal B$ is called \\textit{separating} when for each pair of maps $f,g\\in \\mathcal A$ satisfying that $f^{-1}(e_G)\\cup g^{-1}(e_G)=X$, it holds that $Hf^{-1}(e_G)\\cup Hg^{-1}(e_G)=Y$. 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