  There is no systematic general procedure by which isomorphism classes of Hopf
algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the
general procedure for classification of isomorphism classes of Hopf algebras
which are extensions of the group algebra $\k C_p$ by ${\k}^G$ where $C_p$ is a
cyclic group of prime order $p$ and ${\k}^G$ is the Hopf algebra dual of $\k
G$, $G$ a finite abelian $p$-group and $\k$ is an algebraically closed field of
characteristic $0$. We apply the method to calculate the number of isoclasses
of commutative extensions and certain extensions of this kind of dimension $\le
p^4$.
