Let $F_n$ be the free group on $n\geq 2$ 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 $H$ and we define $\tilde H < H$ to be the subgroup of $H$ generated by $\tilde P$.

**Edit:** We impose another condition which is not satisfied in the excellent counterexample by Mark Sapir given below:

- Suppose that the first basis element $x_1$ of $F_n$ is contained in $H$. (This implies $x_1\in \tilde H$)

Any suggestions, references or counterexamples to either of the following questions are welcome:

- Is $\tilde H=H$ ? If this not the case:
- Is $\tilde H$ of finite index in $H$ ? If not:
- Is the image of $\tilde H$ in $H^{ab}=H / [H,H]$ a subgroup of finite index?