Neutron skin thickness of $^{208}$Pb, $^{116,120,124}$Sn, and $^{40}$Ca determined from reaction cross sections of $^{4}$He scattering

[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.