On sharply edge-transitive permutation groups

We consider the problem of determining the maximum possible out-degree d(n) of a digraph on n vertices which admits a sharply edge-transitive group. We show that d(n) ≥ cn log log n for every n, while d(n) = 1 2n infinitely often. Also, d(n) = n - 1 if and only if n is a prime power, whereas for non-prime-power values of n, we show that n - d(n) tends to infinitely with n. The question has interesting group-theoretic aspects. This and related problems generalise the existence question for projective planes.