Automorphisms of $\mathscr{P}(\lambda)/\mathscr{I}_\kappa$

We study conditions on automorphisms of Boolean algebras of the form $P(\lambda)/I_\kappa$ (where $\lambda$ is an uncountable cardinal and $I_\kappa$ is the ideal of sets of cardinality less than $\kappa$) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every cardinality-preserving automorphism of $P(2^\kappa)/I_{\kappa^+}$ which is trivial on all sets of cardinality $\kappa^+$ is trivial, and that $MA_{\aleph_1}$ implies that every automorphism of $P(\mathbb{R})/Fin$ is trivial on a cocountable set.