${\text{Perm}}(G)$. The verification that it is normalized by $ \lambda(G) $ is very similar to the verification in , using the fact that $\rho(G)$ and $\lambda(G)$ commute inside $ {\text{Perm}}(G) $. To show that the six choices of the pair $ s,t $ yield distinct groups, note that for each such pair the group $ A_{s,t} $ is the only one that contains $ \lambda(s) $ and $ \rho(t) $. Hence, by , the groups $A_{s,t}$ are the underlying groups of the 6 Hopf Galois structures of type $C_4\times C_2$.

The subgroups corresponding to the Hopf-Galois structures of type $ C_{8} $ cannot be described in terms of combinations of elements from $ \lambda(G) $ and $ \rho(G) $, since the order of any such element is at most $ 4 $.

\[lemma\_Cst\] Let $s,t\in\{\sigma,\tau,\sigma\tau\} $ with $ s\neq t$ and let $C_{s,t}$ be generated by the permutation $\eta_{s,t} $ defined in cycle notation by $$\eta_{s,t} = (1\,\,s\,\,t\,\,(st)^{-1}\,\,s^2\,\,s^{-1}\,\,t^{-1}\,\,(st)).$$ Then $C_{s,t}$ is a regular subgroup of ${\text{Perm}}(G)$ that is normalized by $\lambda(G)$ and isomorphic to $C_8$. The 6 choices of the pair $ s,t $ yield distinct groups, and these are the underlying groups of the 6 structures of type $C_8 