  In this paper we compute extension groups in the category of strict
polynomial superfunctors and thereby exhibit certain "universal extension
classes" for the general linear supergroup. Some of these classes restrict to
the universal extension classes for the general linear group exhibited by
Friedlander and Suslin, while others arise from purely super phenomena. We then
use these extension classes to show that the cohomology ring of a finite
supergroup scheme---equivalently of a finite-dimensional cocommutative Hopf
superalgebra---over a field is a finitely-generated algebra. Implications for
the rational cohomology of the general linear supergroup are also discussed.
