The $^{6}\mathrm{Li}$${(}^{7}$Li${,}^{7}$Be${)}^{6}$He charge-exchange reaction leading to the neutron-halo nucleus $^{6}\mathrm{He}$ has been studied at E${(}^{7}$Li) = 350 MeV. Magnetic analysis was used to observe transitions to the known ${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$ = ${0}^{+}$ ground state and the ${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$ = ${2}^{+}$ state at ${\mathit{E}}_{\mathit{x}}$ = 1.8 MeV as well as pronounced resonances at \ensuremath{\sim}5.6 MeV, \ensuremath{\sim}14.6 MeV, and \ensuremath{\sim}23.3 MeV. Coincidences with 430-keV Doppler-shifted \ensuremath{\gamma} rays from the deexcitation in flight of the ${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$ = 1/${2}^{\mathrm{\ensuremath{-}}}$ first-excited state in $^{7}\mathrm{Be}$ were measured to permit the identification of spin-flip transitions. All observed transitions appear to have spin-flip characteristics. The shapes of the experimental angular distributions from ${\mathrm{\ensuremath{\theta}}}_{\mathrm{c}.\mathrm{m}.}$ = 0\ifmmode^\circ\else\textdegree\fi{} to 18\ifmmode^\circ\else\textdegree\fi{} are well described by microscopic one-step finite-range distorted-wave calculations with theoretical shell-model transition amplitudes. For the two low-lying shell-model states the absolute cross sections are also well described. The internal structures of the projectile and ejectile are taken into consideration. A large number of contributions is permitted by the angular momentum couplings. Only the ground state of $^{6}\mathrm{He}$ carries significant Gamow-Teller strength B(GT). Contributions with higher L values from the central spin flip and the tensor interactions ${\mathit{V}}_{\mathrm{\ensuremath{\sigma}}\mathrm{\ensuremath{\tau}}}$ and ${\mathit{V}}_{\mathit{T}\mathrm{\ensuremath{\tau}}}$ are responsible for the mostly structureless distributions observed, and the 0\ifmmode^\circ\else\textdegree\fi{} cross sections are not proportional to B(GT). The strong resonances at \ensuremath{\sim}5.6 MeV and \ensuremath{\sim}14.6 MeV are interpreted as ${2}^{+}$ and (1,2${)}^{\mathrm{\ensuremath{-}}}$ resonances, respectively, with cross sections stronger than predicted presumably due to mixing with continuum states leading to quadrupole and dipole enhancements. It appears that the resonance at \ensuremath{\sim}5.6 MeV does not represent a soft dipole mode originally predicted at ${\mathit{E}}_{\mathit{x}}$=4--7 MeV. \textcopyright{} 1996 The American Physical Society.