Let $F_n$ be the free group on $n$ generators and let $H < F_n$ be a finite index normal subgroup. Let $P\subset F$ be the subset consisting of primitive elements (an element of $F_n$ is called primitive if it is a member of a free basis). Furthermore, we define $\tilde P\subset H$ to be the subset consisting of those powers $x^m$, $m\in \mathbb{Z}$, of primitive elements $x\in P$ which actually are elements of $H$.

I would like to know whether $H$ is generated by $\tilde P$ --- at least whether the index of the subgroup of $H$ generated by $\tilde P$ is of finite index.

Any suggestions, references or counterexamples are welcome.