$\require{AMScd}$
Let $\mathcal{O}$ be an (finite) extension of the $p$-adic integers.. Denote by $R$ the power series ring $\mathcal{O} [[X_1, \dots, X_n]]$ and let $I$ be an ideal such that $R/I$ is integral and flat over $\mathcal{O}$.
Assume we have an $\mathcal{O}$-algebra automorphism $\varphi : R/I \to R/I$ of finite order $e$ with $e$ prime to $p$.
Can we always lift $\varphi$ to an automorphism $\phi : R \to R$ of finite ordrer $e$ (i.e. such that \begin{CD} R @>\phi>> R\\ @VVV @VVV\\ R/I @>\varphi>> R/I \end{CD} is commutative ?