  In this paper, we prove the existence of a particular diagonalization for
normal bounded operators defined on subspaces of $L^2(\mathfrak{S})$ where
$\mathfrak{S}$ is a second countable LCA group. The subspaces where the
operators act are invariant under the action of a group $\Gamma$ which is a
semi-direct product of a uniform lattice of $\mathfrak{S}$ with a discrete
group of automorphisms. This class includes the crystal groups which are
important in applications as models for images. The operators are assumed to be
$\Gamma$ preserving. i.e. they commute with the action of $\Gamma$. In
particular we obtain a spectral decomposition for these operators. This
generalizes recent results on shift-preserving operators acting on lattice
invariant subspaces where $\mathfrak{S}$ is the Euclidean space.
