$\textbf{Disclaimer:}$ As suggested in the comments, I have reposted this question on mathoverflow
Suppose $X$ is a variety, then we have the result that $\operatorname{H}^1(X, \mathcal{O}^*_X)=\operatorname{H}^1_{\acute{e}t}(X,\mathbb{G}_m)=\operatorname{Pic}(X).$ Which also holds in finer sites where coherent sheaves satisfy descent.
I know that a generalisation exists in the no-abelian setting of $\operatorname{GL}_n$, but I do not know exactly how to define $\operatorname{H}^1(-)$. Do you just define it as the set of torsors? Do you define it as a limit of $\check{\operatorname{C}}$ech cohomologies taken over every cover?
My best guess would be to follow the second one, because we want to continue to have something like $0\rightarrow\check{\operatorname{H}}^1(\mathcal{U}, G)\rightarrow \operatorname{H}^1(X, G)\rightarrow \check{\operatorname{H}}^0(\mathcal{U}, \underline{\operatorname{H}}^1(G))$ for a cover $\mathcal{U}$ and a non abelian sheaf $G$ (like in the abelian case using $\check{\operatorname{C}}$ech to sheaf spectral sequence). But this is only a guess, I am not sure.
Most of the answers I have found on the internet suggests I read Giraud's "Cohomologie non abelienne", but my lack of knowledge of French makes that very difficult.