  We develop a mechanism for classication of isomorphism types of non-trivial
semisimple Hopf algebras whose group of grouplikes $G(H)$ is abelian of prime
index $p$ which is the smallest prime divisor of $|G(H)|$. We describe
structure of the second cohomology group of extensions of $\k C_p$ by $\k^G$
where $C_p$ is a cyclic group of order $p$ and $G$ a finite abelian group. We
carry out an explicit classification for Hopf algebras of this kind of
dimension $p^4$ for any odd prime $p$. The ground field is algebraically closed
of characteristic $0$.
