Extra-invariance of group actions

Given discrete groups $\mathrm{\Gamma }\subset \mathrm{\Delta }$ we characterize $(\mathrm{\Gamma },\sigma )$-invariant spaces that are also invariant under $\mathrm{\Delta }$. 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 $(\mathrm{\Gamma },\sigma )$-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 $(\mathrm{\Gamma },\sigma )$-invariant space nearest - in the sense of least squares - to a given set of data.