group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A nonabelian bundle gerbe (as studied by Aschieri-Cantini-Jurco) is a model for the Lie groupoid which is the total space of a smooth $AUT(H)$-principal 2-bundle for $AUT(H)$ the Lie 2-group that is the automorphism 2-group of a Lie group $H$.
Specifically, a nonabelian bundle gerbe on a smooth manifold $X$ is given by a surjective submersion $Y \to X$ and an $H$-bibundle $P \to Y\times_X Y$ together with a morphism of $H$-bibundles
that is associative in the evident sense. This construction serves to model pullbacks of Lie 2-groupoids of the form
where on the rght we have the universal principal 2-bundle.
The resulting Lie groupoid $\tilde P$ is an extension of the Cech groupoid $C(Y)$ by $AUT(H)$. This generalizes the case of ordinary bundle gerbes, which are models for $\mathbf{B}U(1)$-principal 2-bundles, for $\mathbf{B}U(1)$ the circle 2-group.
This can all be extended to topological groupoids, and to structure 2-groups given by more general crossed modules than $H\to Aut(H)$.