Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to M$ whose reduction is the identity map $(Spec\ k,m) \mapsto m$. Let $D_n = Spec\ k[\epsilon]/\langle \epsilon^n \rangle$, and $D_\infty = Spec\ k[[\epsilon]]$ be the inverse limit, thought of as the additive formal group.

> What are the obstructions in extending this map $D_2 \times M \to M$ to a formal group action $D_\infty \times M \to M\ ?$ How unique is such an extension?

Motivation: I'm annoyed by the fact that a function on ${\mathbb A}^1_k$ with first derivative zero, i.e. invariant under the $D_2$-action, need not be constant (it could be a $p$th power). When I asked in http://mathoverflow.net/questions/71423/replacement-for-derivations-in-characteristic-p 
for the right condition to replace it, I was told to take all Hasse derivatives, which I am reinterpreting as having a $D_\infty$- not just $D_2$-action. (I prefer the formal group action to the group action, because it restricts to open sets.)