  Let $k$ be a field of characteristic $2$ and let $H$ be a finite group or
group scheme. We show that the negative Tate cohomology ring
$\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring of
the trivial module in a Verdier localization of the stable category of
$kG$-modules for $G$ an extension of $H$. This means in some cases that the
endomorphism of the trivial module is a local ring with infinitely generated
radical with square zero. This stands in stark contrast to some known
calculations in which the endomorphism ring of the trivial module is the degree
zero component of a localization of the cohomology ring of the group.
