  Despite the discovery of thousands of exoplanets, no exomoons have been
detected so far. We test a recently developed method for exomoon search, the
orbital sampling effect (OSE), using the full exoplanet photometry from the
Kepler Space Telescope. The OSE is applied to phase-folded transits, for which
we present a framework to detect false positives, and discuss four candidates
which pass several of our tests. Using numerical simulations, we inject exomoon
signals into real Kepler data and retrieve them, showing that under favorable
conditions, exomoons can be found with Kepler and the OSE method. In addition,
we super-stack a large sample of Kepler planets to search for the average
exomoon OSE and the accompanying increase in noise, the scatter peak. We find a
significant OSE-like signal, which might indicate the presence of moons, for
planets with 35d<P<80d, having an average dip per planet of 6+-2ppm,
corresponding to a moon radius of 2120 +330 -370km for the average star radius
of 1.24R_Sun in this sample.
