  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
