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We describe the automorphism structure of groups $\\mathrm{FSym}(X)\\le G\\le \\mathrm{Sym}(X)$ and use this to state some conditions on $G$ for it to have the $R_\\infty$ property. Our main results are that if $G$ is infinite, torsion, and $\\mathrm{FSym}(X)\\le G\\le \\mathrm{Sym}(X)$, then it has the $R_\\infty$ property. 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