Autohomeomorphisms of pre-images of $\mathbb N^*$

In the study of the Stone-\u{C}ech remainder of the real line a detailed study of the Stone-\u{C}ech 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 $\mathbb M^*$ induces an autohomeomorphism of $\mathbb N^*$. We prove that it is consistent with there being non-trivial autohomeomorphism of $\mathbb N^*$ that those induced by autohomeomorphisms of $\mathbb M^*$ are trivial.