Commutative group stacks and Galois cohomology

"Classically", if we consider an abelian variety $A$ over some number field $k$, we get a $Gal(bar{k}/k)$-module $A(bar{k})$, or equivalently a sheaf of abelian groups on the étale site $text{Spec}(k)_{ét}$. In particular, we can apply the machinery of Galois cohomology.

My question is if there is a similar construction for abelian stacks. If we have commutative group stacks over number field $k$ with a short exact sequence
$$0rightarrow Arightarrow Brightarrow Crightarrow 0$$
is there a universal way of extending the left exact sequence
$$0rightarrow A(k)rightarrow B(k)rightarrow C(k)$$
i.e. something akin to $H^i(G_k,A)$? My thinking is that we have an exact triangle
$$A^flatrightarrow B^flatrightarrow C^flat rightarrow A^flat[1]$$
in $D(text{Spec}(k)$ on which we can apply the global sections functor. Would that give us something relevant?