Radiative nonrecoil nuclear finite size corrections of order $\alpha (Z\alpha {)}^5$ to the hyperfine splitting of S-states in muonic hydrogen

On the basis of quasipotential method in quantum electrodynamics we calculate nuclear finite size radiative corrections of order $\alpha (Z\alpha {)}^5$ to the hyperfine structure of S-wave energy levels in muonic hydrogen and muonic deuterium. For the construction of the particle interaction operator we employ the projection operators on the particle bound states with definite spins. The calculation is performed in the infrared safe Fried-Yennie gauge. Modern experimental data on the electromagnetic form factors of the proton and deuteron are used.