Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and subscripts mean p-adic completions.

Define the Shafarevich-Tate group of $G$, a $G_K$-module, to be: $$\mathrm{III}(G) = Ker\\ \lgroup\\ H^1(G_K, G(O_\bar{K}) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},G(O_\bar{K_\mathfrak{p}})) \rgroup$$

For $G=GL_1$ it is well known to be isomorphic to the class group of $K$. My question is: $$\text{What is }\mathrm{III}(GL_2)?$$

It is not clear that the above defines a group if $G$ is not abelian. The article "Abelianization of the First Galois Cohomology of Reductive Groups", M. Borovoi, shows that if instead of $G(O_\bar{K})$ we put $G$, a reductive group over $K$, then above has in fact a canonical natural abelian group structure. I suspect this remains true if we restrict the group to $O_\bar{K}$.

The same question for $GL_n$, quaternion units, etc. is also very interesting.