We investigate the W2(k)-liftability of singular schemes. We prove constructibility of the locus of W2(k)-liftable schemes in a flat family X→S
Nearby in the stack
. Moreover, we construct an explicit
W2(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≥4
ordinary double points do not admit a
W2(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