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