  JWST provides an unprecedented opportunity for unbiased surveys of
H$\alpha$-emitting galaxies at $z>4$ with the NIRCam wide-field slitless
spectroscopy (WFSS). In this work, we present a census of Ly$\alpha$ escape
fraction ($f_{esc, Ly\alpha}$) of 165 star-forming galaxies at $z=4.9-6.3$
using their H$\alpha$ emission directly measured from FRESCO NIRCam/WFSS data.
We search for Ly$\alpha$ emission of each H$\alpha$-emitting galaxy in VLT/MUSE
data. The overall $f_{esc, Ly\alpha}$ measured by stacking is $f_{esc,
Ly\alpha}$ is $0.090\pm0.006$. We find that $f_{esc, Ly\alpha}$ displays a
strong dependence on the observed UV slope ($\beta_{\rm obs}$) and E(B-V), such
that the bluest galaxies ($\beta_{\rm obs}\sim-2.5$) have the largest escape
fractions ($f_{\rm esc, Ly\alpha}\approx0.6$), indicative of the crucial role
of dust and gas in modulating the escape of Ly$\alpha$ photons. $f_{esc,
Ly\alpha}$ is less well related to other parameters, including the UV
luminosity and stellar mass, and the variation in $f_{esc, Ly\alpha}$ with them
can be explained by their underlying coupling with E(B-V) or $\beta_{\rm obs}$.
Our results suggest a tentative decline in $f_{esc, Ly\alpha}$ at $z\gtrsim 5$,
implying increasing intergalactic medium attenuation towards higher redshift.
Furthermore, the dependence of $f_{esc, Ly\alpha}$ on $\beta_{\rm obs}$ is
proportional to that of the ionizing photon escape fraction ($f_{\rm esc,
LyC}$), indicating the escape of Ly$\alpha$ and ionizing photon may be
regulated by similar physical processes. With $f_{esc, Ly\alpha}$ as a proxy to
$f_{\rm esc, LyC}$, we infer that UV-faint ($M_{\rm UV}>-16$) galaxies
contribute $>70\%$ of the total ionizing emissivity at $z=5-6$. If these
relations hold during the epoch of reionization, UV-faint galaxies can
contribute the majority of UV photon budget to reionize the Universe.
