I appreciate Deligne-Beilinson cohomology as a topological cohomology generalization of de Rham cohomology, which concerns the topological structure of manifolds.
On the other hand, we know that there is Group Cohomology theory, suitable for describing the classifying spaces $BG$ of a group $G$, for example, see ncatlab: group+cohomology and ncatlab: Dijkgraaf-Witten gauge theory.
Is there a group cohomology version of Deligne-Beilinson cohomology, concerning classifying spaces $B^n G$ of a group $G$? What are some key intro References? Thank you in advance.