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\textbf{Background}: Superallowed $0^+ \rightarrow 0^+$ $\beta$ decays ofisospin $T=2$ nuclides can be used to test theoretical isospin symmetrybreaking 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 scalarcurrents using the $\beta-\nu$ angular correlation. Key ingredients include the$Q_{\textrm{EC}}$ value and branching of the superallowed transition and thehalf 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 intoa plastic scintillator surrounded by an array of high-purity germaniumdetectors used to detect $\beta$-delayed $\gamma$ rays. The high-resolution$\gamma$-ray spectrum was analyzed to measure the $\gamma$-ray energies andintensities. \textbf{Results}: The intensity of $^{20}$Mg $\beta$-delayed$\gamma$ rays through the isobaric analog state in $^{20}$Na was measured to be$(1.60 \pm 0.04_{\textrm{stat}} \pm 0.15_{\textrm{syst}} \pm0.15_{\textrm{theo}}) \times 10^{-4}$, where the uncertainties are statistical,systematic, and theoretical, respectively. The $Q_{\textrm{EC}}$ value for thesuperallowed transition was determined to be $4128.7 \pm 2.2$ keV based on themeasured excitation energy of $6498.4 \pm 0.2_{\textrm{stat}} \pm0.4_{\textrm{syst}}$ 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 preciseto match or exceed the sensitivity required for current low-energy tests of thestandard model.

Superallowed $0^+ \rightarrow 0^+$ $\beta$ decay of $T =2$ $^{20}$Mg: $Q_{\textrm{EC}}$ value and $\beta\gamma$ branching