  Let $G$ be a finite group, and assume that $G$ has an automorphism of order
at least $\rho|G|$, with $\rho\in\left(0,1\right)$. 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 $G$. Furthermore, we generalize an example of
Horo\v{s}evski\u{\i} to show that in finite groups, the quotient of the maximum
automorphism order by the maximum automorphism cycle length may be arbitrarily
large.
