Inducing fixed points in the Stone-Čech compactification

If f is an autohomeomorphism of some space X, then βf denotes its Stone-Čech extension to βX. For each n ⩽ ω, we give an example of a first countable, strongly zero-dimensional subparacompact X and a map f such that every point of X has an orbit of size n under f and βf has a fixed point. We give an example of a normal, zero-dimensional X such that f is fixed point-free but βf is not. We note that it is impossible for every point of X to have an orbit of size 3 and βX to have a point with orbit of size 2.