Counting characters in linear group actions

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 a Irr(GV) such that, for a fixed v a V, the restriction of chi to aOE (c) v > is not a multiple of the regular character of aOE (c) v >. These results might be useful in attacking the noncoprime k(GV)-problem.