No. 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.
Geoff Robinson
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