  Despite recent progress in understanding Lyman-alpha (Lya) 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+-0.4%) of
the LAEs individually in the X-rays (S/N>3) and find an average luminosity of
10^{44.31+-0.01} erg/s and an average black hole accretion rate (BHAR) of
0.72+-0.01 Msun/yr, consistent with moderate to high accreting AGN. We detect
120 sources in deep radio data (radio AGN fraction of 3.2%+-0.3%). The global
AGN fraction (8.6%+-0.4%) rises with Lya luminosity and declines with
increasing redshift. For X-ray detected LAEs, Lya luminosities correlate with
the BHARs, suggesting that Lya luminosity becomes an accretion rate indicator.
Most LAEs (93.1+-0.6%) at 2<z<6 have no detectable X-ray emission (BHARs<0.017
Msun/yr). The median star formation rate (SFR) of star-forming LAEs from Lya
and radio luminosities is 7.6^{+6.6}_{-2.8} Msun/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 Lya
luminosity traces BHAR, making them bright in Lya, and another with low SFRs
which remain undetected in even the deepest X-ray stacks but is detected in the
radio stacks.
