  The simplest orientifolds of the WZW models are obtained by gauging a Z_2
symmetry group generated by a combined involution of the target Lie group G and
of the worldsheet. The action of the involution on the target is by a twisted
inversion g \mapsto (\zeta g)^{-1}, where \zeta is an element of the center of
G. It reverses the sign of the Kalb-Ramond torsion field H given by a
bi-invariant closed 3-form on G. The action on the worldsheet reverses its
orientation. An unambiguous definition of Feynman amplitudes of the orientifold
theory requires a choice of a gerbe with curvature H on the target group G,
together with a so-called Jandl structure introduced in hep-th/0512283. More
generally, one may gauge orientifold symmetry groups \Gamma = Z_2 \ltimes Z
that combine the Z_2-action described above with the target symmetry induced by
a subgroup Z of the center of G. To define the orientifold theory in such a
situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We
reduce the study of the existence of such structures and of their inequivalent
choices to a problem in group-\Gamma cohomology that we solve for all simple
simply-connected compact Lie groups G and all orientifold groups \Gamma = Z_2
\ltimes Z.
