  The fraction of Lyman-$\alpha$ emitters among the galaxy population has been
found to increase from $z\sim0$ to $z\sim6$ and drop dramatically at $z>6$.
This drop has been interpreted as an effect of an increasingly neutral
intergalactic medium with increasing redshift, while a LyC escape fraction
evolving with redshift. We report the result of a large VLT/FORS2 program
aiming to confirm spectroscopically a large galaxy sample at $z\geq6$ that has
been selected in several independent fields through the Lyman Break technique.
Combining those data with archival data, we create a large and homogeneous
sample of $z\sim6$ galaxies ($N=127$), complete in terms of Ly$\alpha$
detection at $>95\%$ for EW(Ly$\alpha)\geq25\AA$. We use this sample to derive
a new measurement of the LAE fraction at $z\sim6$ and derive the physical
properties of these galaxies through spectral energy distribution fitting. We
find a median LAE fraction at $z\sim6$ lower than in previous studies. The main
difference between LAEs and non-LAEs is that the latter are significantly
dustier. Using predictions of our SED fitting code accounting for nebular
emission, we find an effective Ly$\alpha$ escape fraction
$f^{eff}_{esc}(Ly\alpha)=0.23^{+0.36}_{-0.17}$ remarkably consistent with the
value derived by comparing UV luminosity function with Ly$\alpha$ luminosity
function. We conclude that the drop in the LAE fraction from $z\sim6$ to $z>6$
is less dramatic than previously found and the effect of an increasing IGM
neutral fraction is possibly observed at $5<z<6$. Based on our derived
$f^{eff}_{esc}(Ly\alpha)$, we find that the IGM has a relatively small impact
on Ly$\alpha$ photon visibility at $z\sim6$, with a lower limit for the IGM
transmission to \lya\ photons, $T_{IGM}\gtrsim0.20$, likely due to the presence
of outflows. [abdridged]
