The reaction $\overline{p}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ has been studied at six incident antiproton momenta (1.63, 1.77, 1.83, 1.88, 1.95, and $2.20 \frac{\mathrm{GeV}}{c}$). The data were taken from a 150 000-picture exposure in the MURA-ANL hydrogen bubble chamber. After examining ambiguities with other final states, a low contamination sample of 12357 events distributed nearly evenly over the six energies was obtained. Cross sections were determined at each energy. The properties of ${\ensuremath{\rho}}^{0}$, ${\ensuremath{\rho}}^{\ifmmode\pm\else\textpm\fi{}}$, ${\ensuremath{\omega}}^{0}$, and ${\ensuremath{\eta}}^{0}$ mesons produced in this reaction are discussed; evidence for production of the $B$ meson is presented. The ${A}_{2}$, $f$, and $g$ mesons are seen, but are not examined in detail. The cross sections of resonance channels are examined using the maximum-likelihood method, with a likelihood function based on Breit-Wigner amplitudes and meson-decay matrix elements. Particular attention is paid to the stability of the results as the detailed form of the fitting functions is varied. The reaction is dominated by the production of the ${\ensuremath{\rho}}^{0}$, ${\ensuremath{\rho}}^{\ifmmode\pm\else\textpm\fi{}}$, and ${\ensuremath{\omega}}^{0}$ mesons, with cross sections of about 50%, 50%, and 20% of the five-pion cross section, respectively. The largest channel is found to be ${\ensuremath{\rho}}^{0}{\ensuremath{\rho}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}$, accounting for over one third of the five-pion events. Cross sections of all resonance channels are examined as a function of energy. The resonance-channel cross sections are compared to the cross sections determined in experiments at other energies. Qualitative features are found to be consistent between experiments. The cross sections show fluctuations between experiments which are larger than statistically expected.