The autohomeomorphism group of the Cech-Stone compactification of the integers

It is shown to be consistent that there is a nontrivial autohomeomorphism of $\beta{\mathbb N} \setminus {\mathbb N}$, yet all such autohomeomorphisms are trivial on a dense $P$-ideal. Furthermore, the cardinality of the autohomeomorphism group of $\beta{\mathbb N} \setminus {\mathbb N}$ can be any regular cardinal between $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$. The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.