  We investigate the $W_2(k)$-liftability of singular schemes. We prove
constructibility of the locus of $W_2(k)$-liftable schemes in a flat family $X
\to S$. Moreover, we construct an explicit $W_2(k)$-lifting of a Frobenius
split scheme $X$ over a perfect field $k$, reproving Bhatt's existential
result. Furthermore, we study existence of liftings of the Frobenius morphism.
In particular, we prove that in dimension $n \geq 4$ ordinary double points do
not admit a $W_2(k)$-lifting compatible with Frobenius, and that canonical
surface singularities are Frobenius liftable. Combined with Bhatt's results,
the latter result implies that the crystalline cohomology groups over $k$ of
surfaces with canonical singularities are not finite dimensional. As a
corollary of our results, we provide a thorough comparison between the notions
of $W_2(k)$-liftability, Frobenius liftability and classical $F$-singularity
types.
