  We consider the combined effects that overshooting and the $^{12}$C($\alpha$,
$\gamma$)$^{16}$O reaction rate have on variable white dwarf stellar models. We
find that carbon-oxygen white dwarf models continue to yield pulsation
signatures of the current experimental $^{12}$C($\alpha$, $\gamma$)$^{16}$O
reaction rate probability distribution function when overshooting is included
in the evolution. These signatures hold because the resonating mantle region,
encompassing $\simeq$\,0.2\,\Msun\ in a typical $\simeq$\,0.6\,\Msun\ white
dwarf model, still undergoes radiative helium burning during the evolution to a
white dwarf. Our specific models show two potential low-order adiabatic
g-modes, $g_2$ and $g_6$, that signalize the $^{12}$C($\alpha$,
$\gamma$)$^{16}$O reaction rate probability distribution function. Both g-mode
signatures induce average relative period shifts of $\Delta P/P = 0.44 \%$ and
$\Delta P/P = 1.33\%$ for $g_2$ and $g_6$ respectively. We find that $g_6$ is a
trapped mode, and the $g_2$ period signature is inversely proportional to the
$^{12}$C($\alpha$, $\gamma$)$^{16}$O reaction rate. The $g_6$ period signature
generally separates the slower and faster reaction rates, and has a maximum
relative period shift of $\Delta P/P = 3.45\%$. We conclude that low-order
g-mode periods from carbon-oxygen white dwarfs may still serve as viable probes
for the $^{12}$C($\alpha$, $\gamma$)$^{16}$O reaction rate probability
distribution function when overshooting is included in the evolution.
