We have confirmed that a spin-orbit interaction in an inverted I${\mathrm{n}}_{0.53}$G${\mathrm{a}}_{0.47}$As/I${\mathrm{n}}_{0.52}$A${\mathrm{l}}_{0.48}$As quantum well can be controlled by applying a gate voltage. This result shows that the spin-orbit interaction of a two-dimensional electron gas depends on the surface electric field. The dominant mechanism for the change in the spin-orbit interaction parameter can be attributed to the Rashba term. This inverted I${\mathrm{n}}_{0.53}$G${\mathrm{a}}_{0.47}$As/I${\mathrm{n}}_{0.52}$A${\mathrm{l}}_{0.48}$As heterostructure is one of the promising materials for the spin-polarized field effect transistor which is proposed by Datta and Das [Appl. Phys. Lett. 56, 665 (1990)].