Let $N$ be a countably infinite set and let $\mathcal P$ denote power set. I get that the automorphisms of $(\mathcal P(N),\subseteq)$ are all induced by permutations of $N$.
But what can be said about automorphisms of $\mathcal P(N)$ mod finite? That is, mod out by the equivalence relation $A\sim B\iff A$ and $B$ differ only finitely.
This is known as Rudin-Shelah problem. Note that, by Stone duality, this is equivalent to determine the self-homeomorphism group of the Stone-Cech boundary of $N$. Notably, consider the group induced by bijections between two cofinite subsets of $N$ (modulo cofinite coincidence). It maps homomorphically injectively into the automorphism group of $A=\mathcal{P}(N)/\mathrm{fin}$. The problem is whether this map $\Phi$ is surjective, i.e., is every automorphism induced by a bijection between two cofinite subsets.
Using basic model theory Rudin (1956) proved that the continuum hypothesis (CH) implies that the automorphism group of $A$ has cardinal $2^{2^{\aleph_0}}$, so $\Phi$ is far from surjective. Shelah (1982) showed the consistency of ZFC + ($\Phi$ is surjective).
See van Douwen's 1990 (posthumous) paper: "The automorphism group of $\mathcal{P}(\omega)/\mathrm{fin}$ need not be simple". In notes of this article, it is mentioned that under CH, this group (thus of cardinal $2^{2^{\aleph_0}}$) is simple, and attributed to Rubin-Štěpánek (reference Handbook of boolean algebras, vol 2, 1989) .
