TITLE: Self-homeomorphism of Stone-Čech boundary with an isolated fixed point
QUESTION [5 upvotes]: $\DeclareMathOperator\bso{\beta^*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some authors denote it $\omega^*$).

Question. Assume ZFC+CH. Does there exist a self-homeomorphism of $\bso$ with an isolated fixed point (i.e., a fixed point that is not a limit of other fixed points)?

Bonus question: if yes: it it true that for every $x\in\bso$ there exist a self-homeomorphism of $\bso$ with $x$ an isolated fixed point?
Other bonus question: what about asking the self-homeomorphism to have order 2?

Contextual remarks:

Say that a self-homeomorphism of $\bso$ is smooth if it is induced by a bijection between two subsets of $\omega$ with finite complement. If $f\in\Homeo(\bso)$ is smooth then the set of fixed points of $f$ is clopen, hence has no isolated point.

Shelah in the early 1980s showed the existence of models of ZFC in which every self-homeomorphism of $\bso$ is smooth (and in particular in which $\#(\Homeo(\bso))=\mathfrak{c}$). Hence in such models the above question has a negative answer.

Rudin proved in the 1950s that under ZFC+CH, $\#(\Homeo(\bso))=2^\mathfrak{c}$ and its action on $\bso$ is not transitive.

If there's a self-homeomorphism with an isolated fixed point $x$, it is easy to modify it to get another one with unique fixed point $x$.

REPLY [5 votes]: As an answer to the bonus question: no, see K. P. Hart and J. Vermeer. Fixed-point sets of autohomeomorphisms of compact F-spaces,
Proceedings of the American Mathematical Society, 123 (1995), 311–314:
every fixed-point set of an autohomeomorphism of $\omega^*$ is a $P$-set.
The same paper also answers the other bonus question:
every $P$-set is the fixed-point set of an involution.
Apply this to a $P$-point.