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Hyperfine splittings arising from the presence of a quadrupolar nucleus at the ""center"" of a molecule belonging to the point group $Td$ or $Oh$ (e.g. $\mathrm{<mrow\; data-mjx-texclass="ORD">189</mrow>}OsO4$ or $\mathrm{<mrow\; data-mjx-texclass="ORD">235</mrow>}UF6)$ are symmetry forbidden to a high degree or approximation. Nevertheless, quadrupole splittings can be induced by either vibrational or rotational distortions of the molecule, i.e. by distortions similar to those responsible for inducing electric dipole moments in $Td$ molecules. Such hyperfine splittings have been observed in several laboratories using laser saturation spectroscopy. In this paper we investigate theoretically the quadrupole splittings induced by excitation of doubly and triply degenerate vibrations in $Td$ adn $Oh$ molecules. We find that much of the vibration-rotation formalism already present in the methane literature can be applied with only minor changes to the induced-quadrouploe-coupling problem, and that rather simple theoretical relationships can be derived between the quadrupole splitting and the tetrahedral or octrahedral vibration-rotation splitting of a given level. Values for the scalar and tensor contribution to the quadrupole coupling constant have been dervied from experimental data for $OsO4$.