<p>The stabilized automorphism group of a dynamical system (<i>X, T</i>) is the group of all self-homeomorphisms of <i>X</i> that commute with some power of <i>T</i>. We study the stabilized automorphism group of minimal and, more generally, certain transitive dynamical systems. Our approach involves utilizing algebraic tools in a novel way to extract information about the rational eigenvalues of these systems from their stabilized automorphism groups. In particular, we prove that if two minimal systems have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues. Using these tools, we also extend Schmieding’s result on the recovery of entropy from the stabilized automorphism group to include irreducible shifts of finite type.</p>