  Let $X$ and $Y$ be proper birational varieties, say with only rational double
points over a perfect field $k$ of positive characteristic. If $X$ lifts to
$W_n(k)$, is it true that $Y$ has the same lifting property? This is true for
smooth surfaces, but we show by example that this is false for smooth varieties
in higher dimension, and for surfaces with canonical singularities. We also
answer a stacky analogue of this question: given a canonical surface $X$ with
minimal resolution $Y$ and stacky resolution $\mathcal{X}$, we characterize
when liftability of $Y$ is equivalent to that of $\mathcal{X}$.
  The main input for our results is a study of how the deformation functor of a
canonical surface singularity compares with the deformation functor of its
minimal resolution. This extends work of Burns and Wahl to positive
characteristic. As a byproduct, we show that Tjurina's vanishing result fails
for every canonical surface singularity in every positive characteristic.
