Autohomeomorphisms of pre-images of ${\mathbb{N}}^{\ast }$

In the study of the Stone-uCech remainder of the real line a detailed study of the Stone-uCech remainder of the space $\mathbb{N}\times [0,1]$, which we denote as $\mathbb{M}$, has often been utilized. Of course the real line can be covered by two closed sets that are each homeomorphic to $\mathbb{M}$. It is known that an autohomeomorphism of ${\mathbb{M}}^{\ast }$ induces an autohomeomorphism of ${\mathbb{N}}^{\ast }$. We prove that it is consistent with there being non-trivial autohomeomorphism of ${\mathbb{N}}^{\ast }$ that those induced by autohomeomorphisms of ${\mathbb{M}}^{\ast }$ are trivial.