Cohomology rings of extended powers and of free infinite loop spaces

We calculate mod-p cohomology of extended powers, and their group completions which are free infinite loop spaces. We consider the cohomology of all extended powers of a space together and identify a Hopf ring structure with divided powers within which cup product structure is more readily computable than on its own. We build on our previous calculations of cohomology of symmetric groups, which are the cohomology of extended powers of a point, the well-known calculation of homology, and new results on cohomology of symmetric groups with coefficients in the sign representation. We then use this framework to understand cohomology rings of related spaces such as infinite extended powers and free infinite loop spaces.