Shintani (1976) gave a correspondence between the irreducible complex characters of the finite general linear group GL(n, F_q) and the Frobenius-invariant irreducible complex characters of the general linear group GL(n, F_{q^m}) over a finite extension F_{q^m} of F_q. Shintani gave this correspondence through an identity of character values, but did not give an explicit map in terms of a parameterization of the characters. In 2008, Silberger and Zink gave such an explicit description of the Shintani lifting using Zelevinsky's Hopf algebra structure of the characters of the finite general linear group. Kawanaka (1977) gave a correspondence, generalizing that of Shintani's, from the irreducible complex characters of the finite unitary group U(n, F_{q^2}) to the Frobenius-invariant irreducible complex characters of GL(n, F_{q^m}) if m is even, or of U(n, F_{q^{2m}}) if m is odd, again through an identity of character values. This talk will give an explicit description of this lift from U(n, F_{q^2}) to GL(n, F_{q^m}), m even, extending the work of Silberger and Zink. These results are joint work with N. Thiem of University of Colorado at Boulder.