Fixed subgroups of automorphisms of hyperbolic 3-manifold groups

For fixed subgroups $Fix(φ)$ of automorphisms $φ$ on hyperbolic 3-manifold groups $π_{1}(M)$, we observed that $\text{rk}(Fix(φ))<2\text{rk}(π_{1}(M))$ and the constant 2 in the inequality is sharp; we also classify all possible groups $Fix(φ)$.