  Let $\psi$ be a permutation of a finite set $X$. We define $\lambda(\psi)$ to
be the largest fraction of elements of $X$ lying on a single cycle of $\psi$.
For a finite group $G$, we define $\lambda(G)$ to be the maximum among the
values $\lambda(\alpha)$, where $\alpha$ runs through the automorphisms of $G$.
In this paper, we develop tools to deal with questions related to
$\lambda$-values of finite groups and of their automorphisms. As a consequence,
we will be able to give a classification, up to a natural notion of
isomorphism, of those pairs $(G,\alpha)$ where $G$ is a finite group, $\alpha$
is an automorphism of $G$ and $\lambda(\alpha)\geq\frac{1}{2}$.
