  This is the first of a series of papers on a systematic application of the
NRQED bound state theory of Caswell and Lepage to higher-order radiative
corrections to the hyperfine structure of the muonium ground state. This paper
describes the calculation of the $\alpha^2 (Z\alpha)$ radiative correction. Our
result for the complete $\alpha^2 (Z\alpha)$ correction is 0.424(4) kHz, which
reduces the theoretical uncertainty significantly. The remaining uncertainty is
dominated by that of the numerical evaluation of the nonlogarithmic part of the
$\alpha (Z\alpha )^2$ term and logarithmic terms of order $\alpha^4$.
