satisfying $ s \neq t $ and $ s \neq t^{\prime} $ the map $ f: A_{s,t}\rightarrow A_{s,t^{\prime}} $ defined by $$\begin{aligned}
f:&\left\{\begin{array}{rcl}
\lambda(s) & \mapsto & \lambda(s)\\
\rho(t) & \mapsto & \rho(t^{\prime}).
\end{array}\right.\end{aligned}$$ is a $G$-equivariant isomorphism:

With a nearly identical argument we now give the result for Hopf-Galois structures of type $C_8$.

\[3.3\] Let $s, s^{\prime},t,t^{\prime}\in\{\sigma,\tau,\sigma\tau\}$ with $ s \neq t $ and $ s^{\prime} \neq t^{\prime} $. We have $L[C_{s,t}]^G\cong L[C_{s^{\prime},t^{\prime}}]^G$ as Hopf algebras if and only if $ t=t^{\prime} $.

Recall the definition of $ C_{s,t} $ from . The nontrivial $G$-orbits of $C_{s,t}$ are $\{\eta_{s,t},\eta_{s,t}^{3}\}$, $\{\eta_{s,t}^{2},\eta_{s,t}^{6}\}$ and $\{\eta_{s,t}^{5},\eta_{s,t}^{7}\}$, all with stabiliser $\langle t \rangle$. Therefore if $ t \neq t^{\prime} $ then there cannot be a $ G $-equivariant isomorphism between $ C_{s,t} $ and $ C_{s^{\prime},t^{\prime}} $ for any choices of $ s,s^{\prime} $. For fixed $ t $ and $ s,s^{\prime} $ satisfying $ s \neq t $ and $ s^{\prime} \neq t $ let $\eta_{s,t}$ and $\eta_{s^{\prime},t}$ be generators of $C_{s,t}$ and $C_{s^{\prime},t}$ respectively; then the map $ f: C_{s,t}\rightarrow C_{s^{\prime},t} $ defined by $$\begin{aligned}
f:\,&\eta_{s,t}\mapsto\eta_{s^{\prime},t}.\end{aligned}$$ is a $G$-equivariant isomorphism.

The result for the Hopf-Galois structures of type $ Q_{8} $ is 