Nontrivial automorphisms of $\mathcal{P}(\omega )\mathrm{/}\mathrm{F}\mathrm{i}\mathrm{n}$ in Cohen models

We show that if $\kappa <{\mathrm{\aleph }}_{\omega }$ Cohen reals are added to a model of $\mathsf{C}\mathsf{H}$, then there are nontrivial automorphisms of $\mathcal{P}(\omega )\mathrm{/}\mathrm{F}\mathrm{i}\mathrm{n}$ in the extension. Under some further hypotheses on the ground model, namely the existence of long enough sage Davies trees (which follows from $\mathsf{S}\mathsf{C}\mathsf{H}$ plus ${\mathrm{\square }}_{\lambda }$ for every $\lambda$ with $\mathrm{c}\mathrm{f}(\lambda )=\omega$), we prove the same result for cardinals $\kappa \ge {\mathrm{\aleph }}_{\omega }$ as well. This extends a result a Shelah and Steprāns, who proved the result for $\kappa ={\mathrm{\aleph }}_2$.