  For a second countable locally compact group $G$ and a closed abelian
subgroup $H$, we give a range function classification of closed subspaces in
$L^2(G)$ invariant under left translation by $H$. For a family $\mathscr{A}
\subset L^2(G)$, this classification ties with a set of conditions under which
the translations of $\mathscr{A}$ by $H$ form a continuous frame or a Riesz
sequence. When $G$ is abelian, our work relies on a fiberization map; for the
more general case, we introduce an analogue of the Zak transform. Both
transformations intertwine translation with modulation, and both rely on a new
group-theoretic tool: for a closed subgroup $\Gamma \subset G$, we produce a
measure on the space $\Gamma \backslash G$ of right cosets that gives a measure
space isomorphism $G \cong \Gamma \times \Gamma \backslash G$. Outside of the
group setting, we consider a more general problem: for a measure space $X$ and
a Hilbert space $\mathcal{H}$, we investigate conditions under which a family
of functions in $L^2(X;\mathcal{H})$ multiplies with a basis-like system in
$L^2(X)$ to produce a continuous frame or a Riesz sequence in
$L^2(X;\mathcal{H})$. Finally, we explore connections with dual integrable
representations of LCA groups, as introduced by Hern{\'a}ndez et al.
