We investigate the collective coordinate quantization of the icosahedrally symmetric B=7 Skyrmion, which is known to have a ground state with spin 7/2 and isospin 1/2. We find a particular quantum state maximally preserving the symmetries of the classical solution, and also present a novel relationship between the quantum state and the rational map approximation to the classical solution. We also investigate the allowed spin states if the icosahedral symmetry is partially broken. Skyrme field configurations with $D_5$ residual symmetry can be quantized with spin 3/2, giving a realistic model for the ground states of the $^7{rm Li}/^7{rm Be}$ isospin doublet
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