  We study the ground-state properties of a quantum "sunburst model", composed
of a quantum Ising spin-ring in a transverse field, symmetrically coupled to a
set of ancillary isolated qubits, to maintain a residual translation invariance
and also a $\mathbb{Z}_2$ symmetry. The large-size limit is taken in two
different ways: either by keeping the distance between any two neighboring
ancillary qubits fixed, or by fixing their number while increasing the ring
size. Substantially different regimes emerge, depending on the various
Hamiltonian parameters: for small energy scale $\delta$ of the ancillary
subsystem and small ring-qubits interaction $\kappa$, we observe rapid and
nonanalytic changes in proximity of the quantum transitions of the Ising ring,
both first-order and continuous, which can be carefully controlled by
exploiting renormalization-group and finite-size scaling frameworks. Smoother
behaviors are instead observed when keeping $\delta>0$ fixed and in the Ising
disordered phase. The effect of an increasing number $n$ of ancillary spins
turns out to scale proportionally to $\sqrt{n}$ for sufficiently large values
of $n$.
