Following Automorphisms of the Selberg class, I define **strong** automorphisms of the Selberg class by adding as an hypothesis that every invariant of $F$ (i.e all the $H$-invariants, conductor and root number, not only the degree) is preserved under the action of such a strong automorphism. It follows that (see http://www.math.ubc.ca/~gerg/teaching/613-Winter2011/SelbergClass.pdf), if $\Phi$ is a strong automorphism of $\mathcal{S}$, then for every $F\in\mathcal{S}$, $F$ and $\Phi(F)$ have the same functional equation.

My question is: can there be other strong automorphisms of the Selberg class than the identity map and the complex conjugation?

Thanks in advance.