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You can simply use conjugation
sage: S5 = SymmetricGroup(5)
sage: S10 = SymmetricGroup(10)
sage: t = S10("(1,6)(2,7)(3,8)(4,9)(5,10)")
sage: s = S5("(1,2,3)(4,5)")
sage: t*s*~t
(6,7,8)(9,10)
Or be more direct with
sage: f = lambda s: S10(range(1,6) + [s(i)+5 for i in range(1,6)])
sage: f(s)
(6,7,8)(9,10)
