  We study the classical double copy for ungauged half-maximal supergravities
using the Kaluza-Klein reduction of double field theory (DFT). We construct a
general formula for the Kaluza-Klein (KK) reduction of the DFT Kerr-Schild
ansatz. The KK reduction of the ansatz is highly nonlinear, but the associated
equations of motion are linear. This linear structure implies that half-maximal
supergravities admit a classical double copy. We show that their single copy is
given by a pair of Maxwell-scalar theories, which are the KK reduction of a
higher-dimensional single copy of DFT. We also investigate their T-duality
transformations - both the Buscher rule and continuous O(D,D) rotations.
Applying the Buscher rule to the Kerr BH, we obtain a solution with a
nontrivial Kalb-Ramond field and dilaton. We also identify the single copy of
Sen's heterotic BH and the chiral null model and show that the chiral null
model is self-dual under T-duality rotations.
