  Let $G$ be a finite group and $V$ be a finite $G$--module. We present upper
bounds for the cardinalities of certain subsets of $\Irr(GV)$, such as the set
of those $\chi\in\Irr(GV)$ such that, for a fixed $v\in V$, the restriction of
$\chi$ to $<v>$ is not a multiple of the regular character of $<v>$. These
results might be useful in attacking the non--coprime $k(GV)$--problem.
