  Ab initio calculations of the Gamow-Teller (GT) matrix elements in the
$\beta$ decays of $^6$He and $^{10}$C and electron captures in $^7$Be are
carried out using both variational and Green's function Monte Carlo wave
functions obtained from the Argonne $v_{18}$ two-nucleon and Illinois-7
three-nucleon interactions, and axial many-body currents derived from either
meson-exchange phenomenology or chiral effective field theory. The agreement
with experimental data is excellent for the electron captures in $^7$Be, while
theory overestimates the $^6$He and $^{10}$C data by $\sim 2\%$ and $\sim
10\%$, respectively. We show that for these systems correlations in the nuclear
wave functions are crucial to explain the data, while many-body currents
increase by $\sim 2$--$3\%$ the one-body GT contributions. These findings
suggest that the longstanding $g_A$-problem, i.e., the systematic
overprediction ($\sim 20 \%$ in $A\le 18$ nuclei) of GT matrix elements in
shell-model calculations, may be resolved, at least partially, by correlation
effects.
