  Given discrete groups $\Gamma \subset \Delta$ we characterize
$(\Gamma,\sigma)$-invariant spaces that are also invariant under $\Delta$. This
will be done in terms of subspaces that we define using an appropriate Zak
transform and a particular partition of the underlying group. On the way, we
obtain a new characterization of principal $(\Gamma,\sigma)$-invariant spaces
in terms of the Zak transform of its generator. This result is in the spirit of
the analogous in the context of shift-invariant spaces in terms of the Fourier
transform, which is very well-known. As a consequence of our results, we give a
solution for the problem of finding the $(\Gamma,\sigma)$-invariant space
nearest - in the sense of least squares - to a given set of data.
