  Employing internal quantum systems as reference frames is a crucial concept
in quantum gravity, gauge theories and quantum foundations whenever external
relata are unavailable. In this work, we give a comprehensive and
self-contained treatment of such quantum reference frames (QRFs) for the case
when the underlying configuration space is a finite Abelian group,
significantly extending our previous work (Quantum 5, 530 (2021)). The
simplicity of this setup admits a fully rigorous quantum information-theoretic
analysis, while maintaining sufficient structure for exploring many of the
conceptual and structural questions also pertinent to more complicated setups.
We exploit this to derive several important structures of constraint
quantization with quantum information-theoretic methods and to reveal the
relation between different approaches to QRF covariance. In particular, we
characterize the "physical Hilbert space" -- the arena of the
"perspective-neutral" approach -- as the maximal subspace that admits
frame-independent descriptions of purifications of states. We then demonstrate
the kinematical equivalence and, surprising, dynamical inequivalence of the
"perspective-neutral" and the "alignability" approach to QRFs. While the former
admits unitaries generating transitions between arbitrary subsystem relations,
the latter, remarkably, admits no such dynamics when requiring
symmetry-preservation. We illustrate these findings by example of interacting
discrete particles, including how dynamics can be described "relative to one of
the subsystems".
