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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 $G$. Inthis paper, we are interested in the proportion $$\frac{|\{\omega\in \Omega\mid\omega \textrm{ lies in a }G_\alpha\textrm{-orbit of cardinality at mosttwo}\}|}{|\Omega|},$$ that is, the proportion of elements of $\Omega$ lying ina suborbit of cardinality at most two. We show that, if this proportion isgreater than $5/6$, then each element of $\Omega$ lies in a suborbit ofcardinality at most two and hence $G$ is classified by a result of Bergman andLenstra. We also classify the permutation groups attaining the bound $5/6$.  We use these results to answer a question concerning the enumeration ofCayley 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.

Finite transitive groups having many suborbits of cardinality at most two and an application to the enumeration of Cayley graphs