  A flux formulation of Double Field Theory on group manifold is derived and
applied to study generalized Scherk-Schwarz compactifications, which give rise
to a bosonic subsector of half-maximal, electrically gauged supergravities. In
contrast to the flux formulation of original DFT, the covariant fluxes split
into a fluctuation and a background part. The latter is connected to a
$2D$-dimensional, pseudo Riemannian manifold, which is isomorphic to a Lie
group embedded into O($D,D$). All fields and parameters of generalized
diffeomorphisms are supported on this manifold, whose metric is spanned by the
background vielbein $E_A{}^I \in$ GL($2D$). This vielbein takes the role of the
twist in conventional generalized Scherk-Schwarz compactifications. By doing
so, it solves the long standing problem of constructing an appropriate twist
for each solution of the embedding tensor. Using the geometric structure,
absent in original DFT, $E_A{}^I$ is identified with the left invariant
Maurer-Cartan form on the group manifold, in the same way as it is done in
geometric Scherk-Schwarz reductions. We show in detail how the Maurer-Cartan
form for semisimple and solvable Lie groups is constructed starting from the
Lie algebra. For all compact embeddings in O($3,3$), we calculate $E_A{}^I$.
