The fixed subgroups of homeomorphisms of Seifert manifolds

Let $M$ be a compact connected orientable Seifert manifold with hyperbolic orbifold $B_M$, and $f_{\pi}: \pi_1(M)\rightarrow\pi_1(M)$ be an automorphism induced by an orientation-reversing homeomorphism $f$ of $M$. We give a bound on the rank of the fixed subgroup of $f_{\pi}$, namely, $\rank\fix(f_{\pi})<2\rank \pi_1(M)$, which is similar to the inequalities on surface groups and hyperbolic 3-manifold groups.