  We establish a McKay correspondence for finite and linearly reductive
subgroup schemes of $\mathrm{SL}_2$ in positive characteristic. As an
application, we obtain a McKay correspondence for all rational double point
singularities in characteristic $p\geq7$.
  We discuss linearly reductive quotient singularities and canonical lifts over
the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions
of singularities of these canonical lifts via $G$-Hilbert schemes.
  In the appendix, we discuss several approaches towards the notion of
conjugacy classes for finite group schemes: This is an ingredient in McKay
correspondences, but also of independent interest.
