\lambda(s)\rho(t)&\mapsto& -1\\
\lambda(s^2)&\mapsto& 1\\
\lambda(t)\rho(st)&\mapsto& 1\\
\end{array}\right.,\quad
\chi_2:\left\{
\begin{array}{rcl}
\lambda(s)\rho(t)&\mapsto& 1\\
\lambda(s^2)&\mapsto& -1\\
\lambda(t)\rho(st)&\mapsto& 1\\
\end{array}\right.,\quad
\chi_3:\left\{
\begin{array}{rcl}
\lambda(s)\rho(t)&\mapsto& 1\\
\lambda(s^2)&\mapsto& 1\\
\lambda(t)\rho(st)&\mapsto& -1\\
\end{array}\right..\end{aligned}$$ Let $ \chi_{0} $ denote the identity in $ \widehat{E}_{s,t} $, and recall the $G$-orbit structure of $E_{s,t}$ in . It is easily verified that $^s\chi_2=\chi_2\chi_3$, $^t\chi_2=\chi_1\chi_2$ and $^{st}\chi_2=\chi_1\chi_2\chi_3$ and that $s$ and $t$ act trivially on $\chi_0$, $\chi_1$, $\chi_3$ and $\chi_1\chi_3$. Hence the orbits of $ G $ in $\widehat{E}_{s,t} $ are $$\{\chi_0\},\quad \{\chi_1\},\quad \{\chi_3\}, \quad \{\chi_1\chi_3\}, \text{ and } \{\chi_2,\chi_1\chi_2,\chi_2\chi_3,\chi_1\chi_2\chi_3\}.$$ The orbit representatives $ \chi_0, \;  \chi_1, \; \chi_3 \; \chi_1\chi_3 $ all have stabilizer $ G $, and the orbit representative $ \chi_2 $ has stabiliser $ \langle s^2\rangle$. Therefore we have $L[E_{s,t}]^G\cong F^4\times K $, as claimed.

For the remaining structures whose underlying group $ N $ is abelian there may exist characters of $ N $ whose values do not lie in the field $F$. In these cases the action of $ \Omega $ on $ \widehat{N} $ depends upon the intersection of $ L $ with certain cyclotomic extensions of $ F $, and can be difficult to trace in detail. To overcome this problem we study the action 