  Let $G$ be a finite transitive group on a set $\Omega$, let $\alpha\in
\Omega$ and let $G_\alpha$ be the stabilizer of the point $\alpha$ in $G$. In
this paper, we are interested in the proportion $$\frac{|\{\omega\in \Omega\mid
\omega \textrm{ lies in a }G_\alpha\textrm{-orbit of cardinality at most
two}\}|}{|\Omega|},$$ that is, the proportion of elements of $\Omega$ lying in
a suborbit of cardinality at most two. We show that, if this proportion is
greater than $5/6$, then each element of $\Omega$ lies in a suborbit of
cardinality at most two and hence $G$ is classified by a result of Bergman and
Lenstra. We also classify the permutation groups attaining the bound $5/6$.
  We use these results to answer a question concerning the enumeration of
Cayley graphs. Given a transitive group $G$ containing a regular subgroup $R$,
we determine an upper bound on the number of Cayley graphs on $R$ containing
$G$ in their automorphism groups.
