Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S\_W$ defined over a field of char 0. I am interested in properties of this lifting.


For exapmle, I read (but without reference, thus I will appreciate some if one knows) that if I have a K3 of finite height, then there exist a lifting $S\_W$ such that the restirction map of the Picard groups is an isomorphism.
Can I say the same thing for other surfaces?


What about abelian surfaces with a polarization of degree coprime with the characteristic? 


For Enriques surfaces in odd characteristic?


For Bielliptic surfaces in characteristic greater than 3?


Thank you very much for any minute you spend reading this! Double thanks if you also post something :D



Lifting to char 0, references and questions
For an Enriques surface $S$ in odd characteristic, for any lift $S\_W$ over $W$, the restriction map on Picard groups is an isomorphism. Essentially this follows from the fact that $h^2(S,\mathcal{O}\_S)$ equals $0$ together with infinitesimal deformation theory of the Picard functor. I believe this is discussed in Cossec-Dolgachev, also cf. p. 7 of the following survey by Dolgachev,


[A Brief Introduction to Enriques Surfaces](https://www.math.nagoya-u.ac.jp/en/research/conference/2013/download/Kyoto13.pdf)


