Let M=N×[0,1]. The natural projection π:M→N, which sends (n,x) to n, induces a projection mapping π⁎:M⁎→N⁎, where M⁎ and N⁎ denote the Čech-Stone remainders of M and N, respectively. We show that CH implies every autohomeomorphism of N⁎ lifts through the natural projection to an autohomeomorphism of M⁎. That is, for every homeomorphism h:N⁎→N⁎ there is a homeomorphism H:M⁎→M⁎ such that π⁎∘H=h∘π⁎. This complements a recent result of the second author, who showed that this lifting property is not a consequence of ZFC. Combining this lifting theorem with a recent result of the first author, we also prove that CH implies there is an order-reversing autohomeomorphism of H⁎, the Čech-Stone remainder of the half line H=[0,∞).