We have evaluated the electron capture rates on $^{20}$Ne, $^{20}$F, $^{24}$Mg, $^{24}$Na and the $beta$ decay rates for $^{20}$F and $^{24}$Na at temperature and density conditions relevant for the late-evolution stages of stars with $M=8$-12 M$_odot$. The rates are based on recent experimental data and large-scale shell model calculations. We show that the electron capture rates on $^{20}$Ne, $^{24}$Mg and the $^{20}$F, $^{24}$Na $beta$-decay rates are based on data in this astrophysical range, except for the capture rate on $^{20}$Ne, which we predict to have a dominating contribution from the second-forbidden transition between the $^{20}$Ne and $^{20}$F ground states in the density range $log rho Y_e (mathrm{g~cm}^{-3}) = 9.3$-9.6. The dominance of a few individual transitions allows us to present the various rates by analytical expressions at the relevant astrophysical conditions. We also derive the screening corrections to the rates.
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