Three-loop radiative-recoil corrections to hyperfine splitting generated by one-loop fermion factors

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We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by diagrams with one-loop radiative photon insertions both in the electron and muon lines. An analytic result for these nonlogarithmic corrections of order $\ensuremath{\alpha}({Z}^{2}\ensuremath{\alpha})(Z\ensuremath{\alpha})(m/M){\stackrel{\texttildelow{}}{E}}_{F}$ is obtained. This result constitutes a next step in the implementation of the program of reduction of the theoretical uncertainty of hyperfine splitting below 10 Hz.