  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.
