Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$.

Question.What would be an example of a global function field $F$ and a finite Galois extension $E/F$ with non-cyclic Galois group $G=\{1,a,b,ab\}$ of order 4 such that all decomposition groups for $E/F$ arecyclic?

**Motivation.** Let $E/F$ be as above. Consider the following three-dimensional $F$-tori:
\begin{align*}
&T_1=R^1_{E/F} {\Bbb G}_m:=\ker\big[R_{E/F} {\Bbb G}_m \to {\Bbb G}_{m,F}\big],\\
&T_2=(R_{E/F} {\Bbb G}_m)/{\Bbb G}_{m,F}\,.
\end{align*}
Then
$$Ш^1(F,T_1)\cong{\Bbb Z}/2{\Bbb Z},\quad\ Ш^2(F,T_2)\cong {\Bbb Z}/2{\Bbb Z}.$$
See Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.
J. Reine Angew. Math. 327 (1981), 12–80, Example 5.6 and Remark 1.9.4.

Sansuc considers *number fields* $F={\Bbb Q}$ , $E={\Bbb Q}(\sqrt{13},\sqrt{17})$.
I am looking for a similar example with function fields.