A systematic experimental study of inclusive pion double charge exchange in $^{4}\mathrm{He}$ has been undertaken. The reaction $^{4}\mathrm{He}$$({\ensuremath{\pi}}^{+},{\ensuremath{\pi}}^{\ensuremath{-}})4p$ was observed at incident energies of 120, 150, 180, 240, and 270 MeV; the $^{4}\mathrm{He}$$({\ensuremath{\pi}}^{\ensuremath{-}},{\ensuremath{\pi}}^{+})4n$ reaction was observed at incident energies of 180 and 240 MeV. At each incident energy, the doubly differential cross section was measured at three to five outgoing pion laboratory angles between ${25}^{\ifmmode^\circ\else\textdegree\fi{}}$ and ${130}^{\ifmmode^\circ\else\textdegree\fi{}}$. At each angle, cross sections were measured over the range of outgoing pion energies from 10 MeV to the kinematic limit for the reaction in which the final state consists of the oppositely charged pion plus four free nucleons. The spectra of outgoing pions are strikingly different from those observed for the inclusive double charge exchange reaction in heavier nuclei but resemble those observed in the $({\ensuremath{\pi}}^{\ensuremath{-}},{\ensuremath{\pi}}^{+})$ reaction in $^{3}\mathrm{He}$. The forward-angle spectra in the $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ reactions exhibit a prominent peak at high outgoing pion energies. Interpretation of the peaks in $^{3}\mathrm{He}$ ($^{4}\mathrm{He}$) as a three-nucleon (four-nucleon) resonance is ruled out by kinematic analysis. The results of a calculation, wherein the double charge exchange reaction is assumed to proceed as two sequential single charge exchange interactions, suggest that the high-energy peak is naturally explained by this double scattering mechanism. Nonstatic treatment of the $\ensuremath{\pi}N$ interactions and the inclusion of nuclear binding effects appear to be important in reproducing the shape of the energy spectra at forward angles.