TITLE: Nontrivially nontrivial automorphisms of $P(\omega_1)/$fin
QUESTION [14 upvotes]: 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?

REPLY [4 votes]: EDIT:  The following does not work, as pointed out in the comments. I do not know how to make $\varphi$ a homomorphism.

The answer is yes, but for trivial reasons.  Identify $\omega_1$ with $\omega_1\times \omega$.  Every automorphism $\psi$ of $P(\omega)/fin$ induces an automorphism of $P(\omega_1\times \omega)/fin$ as follows:

If $ A=\bigcup_{i\in \omega_1} \{i\}\times A_i$, then let $\varphi(A) = \bigcup_{i\in \omega_1} \{i\}\times \psi(A_i)$.

Note that $\varphi$ is well-defined on $P(\omega_1\times \omega)/fin$, and is nontrivial on any co-countable set.