Extensions of Hopf algebras

We investigate the notion of exact sequences of Hopf algebras. To two Hopf algebras $A$ and $B$, and a data consisting of an action of $B$ on $A$, a cocycle, a coaction of $A$ on $B$, and a co-cocycle we associate a short exact sequence of Hopf algebras $0\to A\to C\to B\to 0$. We define cleft short exact sequences of Hopf algebras and prove that their isomorphism classes are in a bijective correspondence with the quotient set of datas as above such that the cocycle and the co-cocycle are invertible, modulo a natural action of a subgroup of $\mathrm{Reg}(B,A)$.