Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are isomorphic to $\mathbb{P}^1$ over $\bar{K}$, but not over $K$.
Now let $D_\infty$ be the group $\mathbb{G}_m\rtimes\mu_2$ embedded in $PGL_2$ as $\{az:\,a\in \bar{K}^\times\}\cup\{b/z:\,b\in \bar{K}^\times\}$.
My question: Can one give a similar interpretation to the elements of $H^1(G_K,D_\infty)$?
That is: Find a scheme $X$ (I guess it will have $Aut(X)=D_\infty$) together with a nice map (perhaps it is only an injection)
$ \{$$\bar{K}/K$-forms of $X\}\to H^1(G_K,D_\infty)$.
Motivating reference: Silverman, Joseph H.. "The field of definition for dynamical systems on $\mathbb {P}^1$." Compositio Mathematica 98.3 (1995): 269-304.