Summary: The computation of the equivariant surgery obstruction group is very difficult and important, while it inherits all properties of the generalized Grothendieck-Witt group. For the generalized Grothendieck-Witt group which is defined by the ring of integers and the finite group, using Bak-Morimoto’s work, a mapping ring is computed instead of the generalized Grothendieck-Witt group, and the formula of the rank of the mapping ring is obtained. To make the computation easier, the relation of the orbit of the mapping ring and the double coset is given. Finally, using GAP, the rank of Grothendieck-Witt group when the finite group is 5 order symmetry group is computed.