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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 
$$ \begin{align*} \frac{|\{\omega\in \Omega\mid \omega \textrm{ lies in a }G_{\alpha}\textrm{-orbit of cardinality at most 2}\}|}{|\Omega|}, \end{align*} $$
that is, the proportion of elements of 
$\Omega $ lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than 
$5/6$, then each element of 
$\Omega $ lies in a suborbit of cardinality at most 2, 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.
