The phrase twisted cohomology was used by Larmore in
The result is what May and Sigurdsson (see references at twisted cohomology) call a parameterized spectrum, the “parameters” being the points of , which might also be called, in the older topological terminology, an ex-spectrum.
For any map and, for a partial lift of , he constructs a single obstruction class to a full lift
The vanishing of this obstruction is necessary for the existence of a lifting, but it is sufficient only in the usual stable range.
Notice that his cohomology with coefficients in a spectrum does not mean the sequence of cohomology groups with coefficients in the sequence of spaces constituting the spectrum, but rather a single group. He does explore the relation between his single obstruction and the classical obstructions.