  We present a study of seven spectroscopically confirmed (Ly$\alpha$ emitting)
galaxies at redshift $z\simeq6$ using the $JWST$/NIRCam imaging data. These
galaxies, with a wide range of Ly$\alpha$ luminosities, were recently observed
in a series of NIRCam broad- and medium-bands. We constrain the rest-frame
UV/optical continua and measure the H$\alpha$ line emission of the galaxies
using the combination of the $JWST$/NIRCam and archival $HST$/WFC3 infrared
photometry. We further estimate their escape fractions of Ly$\alpha$ photons
($f_{\rm esc}^{\rm Ly\alpha}$) and the production efficiency of ionizing
photons ($\xi_{\rm ion}$). Among the sample, 6/7 galaxies have Ly$\alpha$
escape fractions of ${\lesssim}10\%$, which might be the status for most of
star-forming galaxies at $z\simeq6$. One UV-faint Ly$\alpha$ galaxy with an
extremely blue UV slope owns a large value of $f_{\rm esc}^{\rm Ly\alpha}$
reaching ${\simeq}50\%$. These galaxies spread a broad range of $\xi_{\rm ion}$
over log$_{10}$ $\xi_{\rm ion, 0}$ (Hz erg$^{-1}$) $\sim25.0-26.5$. We find
that UV-fainter galaxies with bluer UV continuum slopes likely have higher
escape fractions of Ly$\alpha$ photons. We also find that galaxies with higher
Ly$\alpha$ line emission tend to produce ionizing photons more efficiently. The
most Ly$\alpha$-luminous galaxy in the sample has a very high $\xi_{\rm ion,
0}$ of log$_{10}$ $\xi_{\rm ion, 0}$ (Hz erg$^{-1}$) $>26$. Our results support
that Ly$\alpha$ galaxies may have served as an important contributor to the
cosmic reionization. Blue and bright Ly$\alpha$ galaxies are excellent targets
for $JWST$ follow-up spectroscopic observations.
