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Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we prove that if $\\rho>1/2$, then $G$ is abelian, and if $\\rho>1/10$, then $G$ is solvable, whereas in general, the assumption implies $[G:\\operatorname{Rad}(G)]\\leq\\rho^{-1.78}$, where $\\operatorname{Rad}(G)$ denotes the solvable radical of $G$. 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