A Counterexample to Wehlau's Conjecture on Noether Numbers

Let $G$ be a finite group, let $V$ be a finite-dimensional $G$-module over a field $k$, and let $U$ be a $G$-submodule of $V$. Wehlau conjectured that the corresponding Noether numbers satisfy $\beta (k[U{]}^G)\le \beta (k[V{]}^G)$. We disprove this conjecture in characteristic $2$. For the dihedral group $D_8$, we construct an inclusion $U\subseteq V$, with ${\dim }_{k}U=5$ and ${\dim }_{k}V=6$, such that $\beta (k[U{]}^{D_8})=6>5=\beta (k[V{]}^{D_8})$. The construction is defined over ${\mathbb{F}}_2$ and remains valid over every field of characteristic $2$. By inflation, it yields counterexamples for every finite group admitting $D_8$ as a quotient.