Fixed subgroups of automorphisms of hyperbolic 3-manifold groups

For fixed subgroups Fix(ϕ) of automorphisms ϕ of hyperbolic 3-manifold groups π1(M), we observe that rk(Fix(ϕ))<2rk(π1(M)) and the constant 2 in the inequality is sharp; we also classify all possible groups Fix(ϕ).