an instance of a well known result (see [@TARP2017 Example 2.4], for example).

\[3.4\] The Hopf algebras $L[\lambda(G)]^G$ and $L[\rho(G)]^G$ are not isomorphic as Hopf algebras.

The $G$-action on $\rho(G)$ is trivial since $\lambda(G)$ and $\rho(G)$ commute. However, the $G$-action on $\lambda(G)$ is conjugation so that the $G$-orbits are the conjugacy classes. Therefore no $G$-equivariant isomorphism can exist.

Finally, we can give the result for the Hopf-Galois structures of type $ D_{4} $.

\[3.5\] The Hopf algebras $L[D_{s,\lambda}]^G$ and $L[D_{s,\rho}]^G$ are pairwise nonisomorphic as Hopf algebras.

Recall the definitions of $ D_{s,\lambda} $ and $ D_{s,\rho} $ from . The non-trivial $G$-orbits of $D_{s,\lambda}$ are $$\{\lambda(s),\lambda(s^{-1})\}, \{\lambda(t)\rho(s),\lambda(t^{-1})\rho(s)\}\text{, and }\{\lambda(st)\rho(s), \lambda((st)^{-1})\rho(s)\},$$ with stabilisers $\langle s\rangle$, $\langle t \rangle$, and $\langle st \rangle$ respectively. If $ s \neq s^{\prime} $ and $ f : D_{s,\lambda} \rightarrow D_{s^{\prime},\lambda} $ is a $ G $-equivariant bijection then by considering stabilisers we see that $ f(\lambda(s)) = \lambda(t^{\prime})\rho(s^{\prime}) $ for some $ t^{\prime} $. But $ \lambda(s) $ has order $ 4 $, whereas $ \lambda(t^{\prime})\rho(s^{\prime}) $ has order $ 2 $. Therefore $ f $ cannot be an isomorphism. The non-trivial $G$-orbits of $D_{s,\rho}$ are $$\{\lambda(s)\rho(t),\lambda(s^{-1})\rho(t)\}\text{ and }\{\lambda(s)\rho(st),\lambda(s^{-1})\rho(st)\}$$ both with stabiliser $\langle s\rangle$. Therefore 