Let $G_i$ be a sequence of $SU(2)$ matrices, where $i=1,2,...,n$; and $P$ represents a permutation of $\left \{ 1,2,...,n \right \}$.
The question is: Does there exist a sequence of $SU(2)$ matrices $W_i$ such that $$W_i^\dagger G_iW_{P(i)}=\varepsilon_ig,$$ where the sign $\varepsilon_i=1$ or $-1$ depending on $i$ while $g\in SU(2)$ does not depend on $i$.
This question is based on some physics problems and I believe the existence of $W_i$, but it seems very difficult to prove it. Thank you very much.
Such matrices don't exist in general.
Consider the case $n=1$, $g=\text{id}$, $\epsilon_1 = +1$, and $G_1$ any matrix different from the identity. Then the only equation we have is $$W_1^\dagger G_1W_1 = \text{id},$$ which is equivalent to $$G_1 = W_1W_1^\dagger = \text{id}$$ (as we asked that $W_1\in SU(2)$), which is impossible.
Maybe this problem has a solution under some stronger assumptions, but I wouldn't know what they could be, and if any further assumption we could make would be useful in the physical situation the problem comes from.