  [Background] We constructed the Kyushu chiral $g$-matrix and confirmed its
reliability at $30 \lesssim E_{\rm in} \lesssim 100 $~MeV and $250 \lesssim
E_{\rm in} \lesssim 400$~MeV for $^{12}$C scattering. Reaction cross section
data of $^{4}$He scattering are available for some nuclides including
$^{208}$Pb. PREX II collaboration reported a thick neutron skin for $^{208}$Pb.
[Purpose] Our purpose is to deduce neutron skin thicknesses of $^{208}$Pb and
some other nuclides from reaction cross sections calculated in terms of the
double folding model with the $g$-matrix. [Methods] We fold the $g$-matrix and
densities given by mean field calculations. In order to remedy
less-constrainedness of the neutron sector, we renormalize densities so as to
reproduce the observed cross sections. [Results] We found that a 3.4~$\%$
renormalization is necessary for $^{208}$Pb. The neutron density obtained from
renormalization results in $R_{\rm skin}=$ 0.416$\pm$0.146 fm by confronting
the precision proton radius. [Conclusions] Our result is consistent with PREX
II and therefore supports larger slope parameter $L$. Results for $^{40}$Ca and
$^{124}$Sn are also consistent with $R_{\rm skin}$ deduced from other
experiments. For $^{116,120}$Sn the present method gives thicker skins.
