  \textbf{Background}: Superallowed $0^+ \rightarrow 0^+$ $\beta$ decays of
isospin $T=2$ nuclides can be used to test theoretical isospin symmetry
breaking corrections applied to extract the CKM matrix element $V_{ud}$ from $T
= 0,1$ decays by measuring precise $ft$ values and also to search for scalar
currents using the $\beta-\nu$ angular correlation. Key ingredients include the
$Q_{\textrm{EC}}$ value and branching of the superallowed transition and the
half life of the parent. \textbf{Purpose}: To determine a precise experimental
$Q_{\textrm{EC}}$ value for the superallowed $0^+ \rightarrow 0^+$ $\beta$
decay of $T=2$ $^{20}$Mg and the intensity of $^{20}$Mg $\beta$-delayed
$\gamma$ rays through the isobaric analog state in $^{20}$Na. \textbf{Method}:
A beam of $^{20}$Mg was produced using the in-flight method and implanted into
a plastic scintillator surrounded by an array of high-purity germanium
detectors used to detect $\beta$-delayed $\gamma$ rays. The high-resolution
$\gamma$-ray spectrum was analyzed to measure the $\gamma$-ray energies and
intensities. \textbf{Results}: The intensity of $^{20}$Mg $\beta$-delayed
$\gamma$ rays through the isobaric analog state in $^{20}$Na was measured to be
$(1.60 \pm 0.04_{\textrm{stat}} \pm 0.15_{\textrm{syst}} \pm
0.15_{\textrm{theo}}) \times 10^{-4}$, where the uncertainties are statistical,
systematic, and theoretical, respectively. The $Q_{\textrm{EC}}$ value for the
superallowed transition was determined to be $4128.7 \pm 2.2$ keV based on the
measured excitation energy of $6498.4 \pm 0.2_{\textrm{stat}} \pm
0.4_{\textrm{syst}}$ keV and literature values for the ground-state masses of
$^{20}$Na and $^{20}$Mg. \textbf{Conclusions}: The $\beta$-delayed
$\gamma$-decay branch and $Q_{\textrm{EC}}$ value are now sufficiently precise
to match or exceed the sensitivity required for current low-energy tests of the
standard model.
