The $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ reaction cross section as a function of the incident energy is calculated using a model that is an extension of our recently published model of the $e^{+}{e}^{-}$ annihilation into four charged pions. The latter considered the intermediate states with the $\pi$, $\rho$, and $a_1$ mesons and fixed the mixing angle of the $a_1\rho \pi$ Lagrangian and other parameters by fitting the cross section data. Here we supplement the original intermediate states with those containing $\omega (782)$ and $h_1(1170)$, but keep unchanged the values of those parameters that enter both charged and mixed channel calculations. The inclusion of $\omega$ is vital for obtaining a good fit to the cross section data, while the intermediate states with $h_1$ further improve it. Finally, we merge our models of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ and $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ reactions and obtain a simultaneous good fit.

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