Velickovic proved (Theorem 4.1 of OCA and automorphisms of $\mathcal{P}(\omega)/\mathrm{fin}$) that, assuming

- OCA (Open Coloring Axiom) and
- $\rm MA_{\aleph_1}$,

every (Boolean algebra) automorphism of $\mathcal{P}(\omega_1)/\mathrm{ fin}$ is *trivial*, i.e. for every such automorphism $\varphi$ there is a function $e : \omega_1\to \omega_1$ such that for all $a\subseteq \omega_1$, $\varphi[a] = [e''a]$.

On the other hand, if there is a nontrivial automorphism $\psi$ of $\mathcal{P}(\omega)/\mathrm{fin}$ (as there are in any model of CH), one can easily construct a nontrivial automorphism $\varphi$ of $\mathcal{P}(\omega_1)/\mathrm{fin}$ by just copying $\psi$ on $\omega$ and the identity on $\omega_1\setminus \omega$:

$$ \varphi[x] = \psi[x\cap \omega]\vee [x\setminus \omega] $$

Of course one can replace $\omega$ with any countable set $a$, and the identity with any trivial automorphism of $\mathcal{P}(\omega_1 \setminus a)/\mathrm{fin}$. But this is somewhat unsatisfying; all of these automorphisms seem to be nontrivial "for trivial reasons." Hence the following question:

Is there, consistently, an automorphism of $\mathcal{P}(\omega_1)/\mathrm{fin}$ which is nontrivial on every cocountable set?