  Briefly: Using a novel $(1,1)$ superspace formulation of semichiral sigma
models with $4D$ target space, we investigate if an extended supersymmetry in
terms of semichirals is compatible with having a $4D$ target space with
torsion.
  In more detail: Semichiral sigma models have $(2,2)$ supersymmetry and
Generalized K\"ahler target space geometry by construction. They can also
support $(4,4)$ supersymmetry and Generalized Hyperk\"ahler geometry, but when
the target space is four dimensional indications are that the geometry is
restricted to Hyperk\"ahler. To investigate this further, we reduce the model
to $(1,1)$ superspace and construct the extra (on-shell) supersymmetries there.
We then find the conditions for a lift to $(2,2)$ super space and semichiral
fields to exist. Those conditions are shown to hold for Hyperk\"ahler
geometries. The $SU(2)\otimes U(1)$ WZW model, which has $(4,4)$ supersymmetry
and a semichiral description, is also investigated. The additional
supersymmetries are found in $(1,1)$ superspace but shown {\em not} to be
liftable to a $(2,2)$ semichiral formulation.
