  We study the ramification on the cohomology of a smooth proper surface $X$ in
mixed characteristic, in the particular case where $X$ degenerates to a surface
over $\overline{\mathbb{F}}_p$ with simple singularities, also known as
rational double points. We find that the associated monodromy action of inertia
depends on a formal affine neighborhood of the singularity, and under
sufficient restrictions on characteristic $p$, it is tamely ramified and
generated by a conjugacy class representative of an appropriate Weyl group
related to the singularity. Along the way we extend to mixed characteristic
some results of Brieskorn and Slodowy concerning simultaneous resolutions of
surface singularities. We also compare our Weyl group actions to certain
Springer representations constructed by Borho and MacPherson, via the notion of
relative perversity as developed by Hansen and Scholze.
