  Let G be an LCA group and K a closed subgroup of G. A closed subspace of
L^2(G) is called K-invariant if it is invariant under translations by elements
of K.
  Assume now that H is a countable uniform lattice in G and M is any closed
subgroup of G containing H. In this article we study necessary and sufficient
conditions for an H-invariant space to be M-invariant.
  As a consequence of our results we prove that for each closed subgroup M of G
containing the lattice H, there exists an H-invariant space S that is exactly
M-invariant. That is, S is not invariant under any other subgroup M' containing
M. We also obtain estimates on the support of the Fourier transform of the
generators of the H-invariant space, related to its M-invariance.
