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 $beta- u$ 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.
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