Hamiltonian construction conjecture for Pauli-group extensions of the quaternion group

Let $Q_8$ be the quaternion group, and let $B$ be an abelian group containing at most one element of order $2$. Consider a group of the form

$$A=Q_8\circ B.$$

Here $H_{eff}$ denotes the effective Hamiltonian constructed in Theorem.

Hamiltonian construction conjecture. Groups of this form may admit a construction of the Hamiltonian analogous to the construction of $H_{eff}$, such that passing from the larger group $A$ to the smaller group $Q_8$ does not affect any of the dynamical aspects.

This conjecture proposes that the dynamical information can remain entirely within the quaternion subgroup for a broader class of extensions than the specific Pauli-group example studied in the paper. The source does not provide additional hypotheses or evidence establishing when such a Hamiltonian construction exists.