Generating a group by its $q$-elements.

Question

Let $G=PQ$ be a solvable group where $P$ and $Q$ are $p$-subgroup and $q$-subgroup of $G$ respectively. Also suppose that $Q$ is not normal in $G$. Is it true that the group generated by all $q$-elements of $G$ are equal to $G$ i.e. $G=\langle\bigcup_{g\in G}Q^g\rangle$?