Cross sections for one-electron transfer from a He atom into the fully stripped, hydrogenlike, heliumlike, and lithiumlike ${\mathrm{B}}^{q+}$, ${\mathrm{C}}^{q+}$, ${\mathrm{N}}^{q+}$, ${\mathrm{O}}^{q+}$, ${\mathrm{F}}^{q\ensuremath{-}}$, ${\mathrm{Ne}}^{q+}$ ions, and also highly stripped ${\mathrm{S}}^{q+}$ ions have been measured at the energy range of $1.5q\ensuremath{-}3.0q$ keV. The measured cross sections are nearly independent of the collision energy with a few exceptions, and most of the cross sections measured are about (1\ensuremath{\sim}4)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$, but the cross sections for ${\mathrm{B}}^{4+}$, ${\mathrm{C}}^{4+}$, and ${\mathrm{N}}^{4+}$ ions are very small in the energy range studied. When the cross sections measured are plotted as a function of the ionic charge $q$ of isoelectronic projectile ions, strong oscillations in the cross sections are observed. As a first approximation, this oscillatory behavior can be explained in terms of the classical one-electron model.