  Let $K$ be a finite field extension of $\Q$ and let $N$ be a finite group
with automorphism group $F=\Aut(N)$. R. Haggenm\"{u}ller and B. Pareigis have
shown that there is a bijection \[\Theta: {\mathcal Gal}(K,F)\rightarrow
{\mathcal Hopf}(K[N])\] from the collection of $F$-Galois extensions of $K$ to
the collection of Hopf forms of the group ring $K[N]$. For $N=C_n$, $n\ge 1$,
$C_p^m$, $p$ prime, $m\ge 1$, and $N=D_3,D_4,Q_8$, we show that $\Q[N]$ admits
an absolutely semisimple Hopf form $H$ and find $L$ for which $\Theta (L)=H$.
Moreover, if $H$ is the Hopf algebra given by a Hopf-Galois structure on a
Galois extension $E/K$, we show how to construct the preimage of $H$ under
$\Theta$ assuming certain conditions.
