I thought of this several months ago and forgot about it. Now I rethought of it again and I just can't find it anywhere in the literature, so I'll ask here.
Is it known whether or not there exists a (smooth, proper, ...) variety over a field $k$ (perfect? alg. closed?) of positive characteristic that lifts to characteristic $0$ over some ramified extension of $W(k)$ and also lifts to $W_2(k)$, but does not lift over $W(k)$ itself?
In other words, if it lifts to characteristic $0$ (in a way related to $W(k)$) and it lifts to $W_2(k)$ must it lift via $W(k)$ itself?
I have looked at examples that lift to $W_2(k)$ but not to $W_3(k)$ and hence not over $W(k)$, but they don't seem to lift to char $0$ at all. I've also looked examples that lift over a ramified extension but not $W$, but these can easily be shown to not lift to $W_2$ as well.