to each other as Hopf algebras. That is, $L[E_{\sigma,\tau}]^G\cong L[E_{\tau,\sigma}]^G$ as Hopf algebras.

Recall the definition of $ E_{s,t} $ from . The non-trivial $G$-orbits of $E_{s,t}$ are $\{\lambda(s)\rho(t),\lambda(s^{-1})\rho(t)\}$, $\{\lambda(t)\rho(st),\lambda(t^{-1})\rho(st)\}$ and $\{\lambda(st)\rho(s),\lambda((st)^{-1})\rho(s)\}$ with stabilisers $\langle s\rangle$, $\langle t \rangle$ and $\langle st \rangle$ respectively. The map $ f: E_{s,t}\rightarrow E_{t,s} $ defined by $$\begin{aligned}
f: &\left\{\begin{array}{rcl}
\lambda(s)\rho(t) & \mapsto & \lambda(s)\rho((st)^{-1})\\
\lambda(s^2) & \mapsto & \lambda(s^2)\\
\lambda(t)\rho(st) & \mapsto & \lambda(t)\rho(s).
\end{array}\right.\end{aligned}$$ is a $ G $-equivariant isomorphism.

Now we find that for the Hopf-Galois structures of type $C_4\times C_2$ the Hopf algebra isomorphism classes are determined by the choice of $s$.

\[3.2\] Let $s, s^{\prime},t,t^{\prime}\in\{\sigma,\tau,\sigma\tau\}$ with $ s \neq t $ and $ s^{\prime} \neq t^{\prime} $. We have $L[A_{s,t}]^G\cong L[A_{s^{\prime},t^{\prime}}]^G$ if and only if $ s=s^{\prime} $.

Recall the definition of $ A_{s,t} $ from . The non-trivial $G$-orbits of $A_{s,t}$ (that is, those of length greater than one) are $\{\lambda(s),\lambda(s^{-1})\}$, $\{\lambda(s)\rho(t),\lambda(s^{-1})\rho(t)\}$ both with stabiliser $\langle s \rangle$. Therefore if $ s \neq s^{\prime} $ then there cannot be a $ G $-equivariant isomorphism between $ A_{s,t} $ and $ A_{s^{\prime},t^{\prime}} $ for any choices of $ t,t^{\prime} $. For fixed $ s $ and $ t,t^{\prime} $ 