This work examines the microscopic connection between the fermion dynamical symmetry model (FDSM) and an effective bosonic description of collective nuclear states. A simplified FDSM Hamiltonian containing monopole and quadrupole pairing terms together with proton and neutron multipole interactions is mapped onto an s–d boson space by three procedures: the generalized Dyson mapping, a seniority-based Hermitization, and the Belyaev–Zelevinsky expansion. The role of the physical boson subspace, Hermiticity, and the truncation of the boson expansion to one- and two-body operators is discussed. The Dyson representation preserves the fermionic matrix elements in the appropriate physical subspace but is generally non-Hermitian in the conventional boson metric. The seniority and Belyaev–Zelevinsky procedures provide Hermitian IBM-like Hamiltonians that can be compared directly with the interacting boson model. The mapped Hamiltonians are applied to the even osmium isotopes ¹⁸⁶Os, ¹⁸⁸Os, ¹⁹⁰Os, and ¹⁹²Os. The effective parameters evolve comparatively smoothly from ¹⁸⁶Os to ¹⁹⁰Os, whereas several coefficients show a pronounced rearrangement at ¹⁹²Os. The same mass region is singled out by different mappings, supporting an interpretation in terms of a change in the effective collective structure. The parameter discontinuity alone is not taken as proof of a quantum phase transition; further tests using spectra, B(E2) values, quadrupole observables, and collective energy surfaces are required.