Wreath products and the non-coprime $k(GV)$ problem

Let $G=X\wr H$ be the wreath product of a nontrivial finite group $X$ with $k$ conjugacy classes and a transitive permutation group $H$ of degree $n$ acting on the set of $n$ direct factors of $X^n$. If $H$ is semiprimitive, then $k(G)\le k^n$ for every sufficiently large $n$ or $k$. This result solves a case of the non-coprime $k(GV)$ problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.