Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume that $A$ lifts to $W(k)$. 


When it is possible to lift the Frobenius endomorphism of $A$ to $W(k)$?


Probably, $B$ has to be ordinary as only in this case Frobenius endomorphism of $B$ lifts. My problem is that I cannot use obstruction theory here as the existence of liftings on finite levels does not yield to a lifting on the whole $W(k)$.


Maybe some small examples can be done. The case $T=\mathbb{G}\_m,\dim B=1$ is already very interesting. 



Lifting of Frobenius on semi-abelian varieties
[My answer to your other question](https://mathoverflow.net/a/233455/3847) seems to show that if $B$ is ordinary, $A$ has a natural lift over $W$ as a scheme, together with a lift of the Frobenius. I don't see however why the group structure of $A$ should lift, and why it should be compatible with the lift of Frobenius.


**EDIT.** Regarding the group structure: say $B$ is a $T$-torsor over an abelian $A$, trivialized over $0\in A$. We want to give $B$ a group structure such that $p:B\to A$ is a homomorphism with kernel $T$. This should be a map $b:B\times B\to B\to B$ such that the square
$\require{AMScd}$
\begin{CD}
 B\times B @>b>> B \\
 @V{p\times p}VV @VVpV \\
 A\times A @>a>> A \\
\end{CD}
commutes. This map should factor through a map $b':B\times B\to B' = a^\* B$ from a $T\times T$-torsor $B\times B$ over $A\times A$ to the $T$-torsor $B'$ over $A\times A$, equivariant with respect to the addition map $t:T\times T\to T$. It should be easy to write out explicitly in terms of line bundles corresponding to the torsor $B$ what this map is, and what it should satisfy to give $B$ a group structure, and most probably we will see that this structure is preserved by Teichmueller lifts, so that we get a group structure on the canonical lift. 


**EDIT 2.** Some time ago I was pointed in a similar context to look up "bi-extensions" (SGA 7 I, Exp. VII-VIII), but I never did. Perhaps you will find an answer to your question there.


