Not when $p >5$. Conjugation by $\rho$ induces an automorphism of order $2n$ of $\langle \pi \rangle.$ 
Since $\pi$ commutes with nothing other than its powers, this means that $\rho$ must be an element of order $2n.$ This is $p-1$ when $p =3$ or $p =5,$ but not for bigger Fermat primes.