Lifting vector bundles to Witt vector bundles

Let $X$ be a scheme. Let $r \geq 2$ be an integer. Denote by $W_r(X)$ the scheme of Witt vectors of length $r$, built out of $X$. We are concerned with the question of extending (=lifting) vector bundles on $X$, to vector bundles on $W_r(X)$-promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p>0$, and extend them to reduced $k$-schemes. Some of these results were recently recovered, in another form, by Stefan Schröer. As an application, we prove that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.