Kepler photometry of A stars shows that a considerable fraction (about 19 per cent) have a peculiar feature in the periodogram. This feature consists of a broad peak, thought to be due to differential rotation in a spotted star, and a sharp peak at slightly higher frequency. The pattern clearly involves some widespread stellar property and the sharp peak implies a strictly coherent periodicity. We investigate the possibility that the periodicity is due to rotation, pulsation or an orbital effect. We argue that neither rotation nor pulsation can provide a suitable, testable, explanation. We suggest that the sharp feature could be due to a planet in synchronous orbit around the rapidly rotating, spotted A star, not necessarily in transit. Spectroscopic observations of sufficient precision are required to falsify this hypothesis.