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We introduce a family $\mathbb{M}_{r;f,\eta}$ of infinitesimal supergroupschemes, which we call multiparameter supergroups, that generalize theinfinitesimal Frobenius kernels $\mathbb{G}_{a(r)}$ of the additive groupscheme $\mathbb{G}_{a}$. Then, following the approach of Suslin, Friedlander,and Bendel, we use functor cohomology to define characteristic extensionclasses for the general linear supergroup $GL_{m|n}$, and we calculate howthese classes restrict along homomorphisms $\rho: \mathbb{M}_{r;f,\eta}\rightarrow GL_{m|n}.$ Finally, we apply our calculations to describe (up to afinite surjective morphism) the spectrum of the cohomology ring of the $r$-thFrobenius kernel $GL_{m|n(r)}$ of the general linear supergroup $GL_{m|n}$.

Graded analogues of one-parameter subgroups and applications to the cohomology of $GL_{m|n(r)}$