How different solutions for gauge transformations $G_U$ of a PSG are in one-to-one correspondence with the elements in IGG?

Between eqs. (47) and (48) of Wen’s article "Quantum Orders and Symmetric Spin Liquids"
(https://arxiv.org/abs/cond-mat/0107071) is the statement "Thus for each symmetry transformation $U$, the different choices of $G_U$ have a one to one correspondence with the
elements in IGG."

How does this come about from the previous statement "If IGG is non-trivial, then for a fixed symmetry transformation $U$, there are can be many gauge transformations $G_U$ that leave the ansatz unchanged. If $G_U U$ is in the PSG of $u_{ij}$ , $GG_U U$ will also be in the PSG iff $Gin G$ "?