Despite recent progress in understanding Ly {alpha} emitters (LAEs), relatively little is known regarding their typical black hole activity across cosmic time. Here, we study the X-ray and radio properties of ~4000 LAEs at 2.2<z<6 from the SC4K survey in the COSMOS field. We detect 254 (6.8 per cent +/- 0.4 per cent) LAEs individually in the X-rays (S/N>3) with an average luminosity of 10^44.31+/-0.01^erg/s and average black hole accretion rate (BHAR) of 0.72+/-0.01M_{sun}_/yr, consistent with moderate to high accreting active galactic nuclei (AGNs). We detect 120 sources in deep radio data (radio AGN fraction of 3.2 per cent +/- 0.3 per cent). The global AGN fraction (8.6 per cent +/- 0.4 per cent) rises with Ly {alpha} luminosity and declines with increasing redshift. For X-ray-detected LAEs, Ly {alpha} luminosities correlate with the BHARs, suggesting that Ly {alpha} luminosity becomes a BHAR indicator. Most LAEs (93.1 per cent +/- 0.6 per cent) at 2<z<6 have no detectable X-ray emission (BHARs<0.017M_{sun}_/yr). The median star formation rate (SFR) of star-forming LAEs from Ly {alpha} and radio luminosities is 7.6^+6.6^_-2.8_M_{sun}_/yr. The black hole to galaxy growth ratio (BHAR/SFR) for LAEs is <0.0022, consistent with typical star-forming galaxies and the local BHAR/SFR relation. We conclude that LAEs at 2<z<6 include two different populations: an AGN population, where Ly {alpha} luminosity traces BHAR, and another with low SFRs which remain undetected in even the deepest X-ray stacks but is detected in the radio stacks.