  In this article we study the structure of $\Gamma$-invariant spaces of
$L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is
with respect to the action of $\Gamma$, a non commutative group in the form of
a semidirect product of a discrete cocompact subgroup of $\bf R$ and a group of
automorphisms. This class includes in particular most of the crystallographic
groups. We obtain a complete characterization of $\Gamma$-invariant subspaces
in terms of range functions associated to shift-invariant spaces. We also
define a new notion of range function adapted to the $\Gamma$-invariance and
construct Parseval frames of orbits of some elements in the subspace, under the
group action. These results are then applied to prove the existence and
construction of a $\Gamma$-invariant subspace that best approximates a set of
functional data in $L^2(\bf R)$. This is very relevant in applications since in
the euclidean case, $\Gamma$-invariant subspaces are invariant under rigid
movements, a very sought feature in models for signal processing.
