group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The notion of twisted differential c-structures applied to Wu classes.
See at M5-brane the section Conformal blocks and 7d Chern-Simons dual.
Topological twisted Wu structures are considered in
A smooth and differential refinement is discussed in
A truncation of this reproduces the differential integra Wu structures considered in