The author proves the following theorem. Let \(G\) be a connected Lie group of dimension \(n\geq 2\). Then there exist complete hyperbolic Stein manifolds \(\Omega\), dim\(_{\mathbb{C}}\Omega=n\), such that Aut\((\Omega)=G\). After recalling some terminologies concerning the Grauert tubes and the existence of Stein Grauert tubes, the author proves the subrigidity characterization of certain domains. Then some specific domains and perturbations are constructed explicitly. Next he perturbs domains in the tangent bundle of a connected Lie group in an invariant way such that extra symmetry on each fiber would be eliminated. By constructing such kind of domains in a fairly explicitly way, the realization of a connected Lie group as an automorphism group follows from the subrigidity derived above.