group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
While a permutative category already induces a symmetric spectrum, representing the corresponding K-theory of a permutative category, for a bipermutative category this naturally carries the additional structure of an E-infinity ring spectrum. The generalized (Eilenberg-Steenrod) cohomology represented by this spectrum is called the (algebraic) K-theory of $C$ and it is hence a multiplicative cohomology theory.
Peter May, $E_\infty$ Ring Spaces and $E_\infty$ Ring spectra, Springer lectures notes in mathematics, Vol. 533, (1977) (pdf) chaper VI
Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, K-Theory 0680 (web, pdf)
Anthony Elmendorf, Michael Mandell, Permutative categories, multicategories, and algebraic K-theory (arXiv:0710.0082)