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.



What is the interpretation of this galois cohomology set?
You can take $X={\mathbb G}\_m$ (considered as an affine curve). Then $Aut(X)=D\_{\infty}$
and you will have a desired bijection. In this case it is not difficult to describe the set of all $\bar K/K-$forms of $X$: they are curves $x^2-dy^2=m$. For example for $K={\mathbb R}$ we have 3 such curves up to ${\mathbb R}-$isomorphism: $x^2-y^2=1$ and $x^2+y^2=\pm 1$.


