  We examine the cohomology and representation theory of a family of finite
supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes
(\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a
certain relation holds in the cohomology ring, and deduce that for finite
supergroup schemes having this as a quotient, both cohomology mod nilpotents
and projectivity of modules is detected on proper sub-super\-group schemes.
This special case feeds into the proof of a more general detection theorem for
unipotent finite supergroup schemes, in a separate work of the authors joint
with Iyengar and Krause.
  We also completely determine the cohomology ring in the smallest cases,
namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and
$(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation
uses the local cohomology spectral sequence for group cohomology, which we
describe in the context of finite supergroup schemes.
