ABSTRACT Despite recent progress in understanding Ly α 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 ($6.8{\rm{ per\ cent}} ± 0.4{\rm{ per\ cent}}$) LAEs individually in the X-rays (S/N > 3) with an average luminosity of $\rm 10^{44.31± 0.01}\, erg\, s^{-1}$ and average black hole accretion rate (BHAR) of $\rm 0.72 ± 0.01$ M⊙ yr−1, consistent with moderate to high accreting active galactic neuclei (AGNs). We detect 120 sources in deep radio data (radio AGN fraction of $3.2{\rm{ per\ cent}} ± 0.3{\rm{ per\ cent}}$). The global AGN fraction ($\rm 8.6{\rm{ per\ cent}} ± 0.4{\rm{ per\ cent}}$) rises with Ly α luminosity and declines with increasing redshift. For X-ray-detected LAEs, Ly α luminosities correlate with the BHARs, suggesting that Ly α luminosity becomes a BHAR indicator. Most LAEs ($93.1{\rm{ per\ cent}} ± 0.6{\rm{ per\ cent}}$) at 2 < $z$ < 6 have no detectable X-ray emission (BHARs < 0.017 M⊙ yr−1). The median star formation rate (SFR) of star-forming LAEs from Ly α and radio luminosities is $7.6^{+6.6}_{-2.8}$ M⊙ yr−1. 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 α 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.