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For a second countable locally compact group $G$ and a closed abeliansubgroup $H$, 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 whichthe translations of $\mathscr{A}$ by $H$ form a continuous frame or a Rieszsequence. When $G$ is abelian, our work relies on a fiberization map; for themore general case, we introduce an analogue of the Zak transform. Bothtransformations intertwine translation with modulation, and both rely on a newgroup-theoretic tool: for a closed subgroup $\Gamma \subset G$, we produce ameasure on the space $\Gamma \backslash G$ of right cosets that gives a measurespace isomorphism $G \cong \Gamma \times \Gamma \backslash G$. Outside of thegroup setting, we consider a more general problem: for a measure space $X$ anda Hilbert space $\mathcal{H}$, we investigate conditions under which a familyof 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 integrablerepresentations of LCA groups, as introduced by Hern{\'a}ndez et al.

Subspaces of $L^2(G)$ invariant under translation by an abelian subgroup