We compute the mod-2 cohomology of the collection of all symmetric groups as
a Hopf ring, using the transfer product of Strickland and Turner, which sheds
considerable light on the cup product structure of the cohomology of an
individual symmetric group. The main ingredient is a primitivity result for the
coproduct on homology dual to the transfer product. We also briefly develop
related Hopf ring structures on rings of symmetric invariants. Our primary
generating set consists of classes which are linearly dual to homology classes
in Nakaoka's basis, but we also develop a generating set consisting of
Stiefel-Whitney classes of regular representations.