Yes, of course. My question is there any classification of such $s$. For such $s$ it is necessary that $W(s)$ is larger then $W(\tilde{s})$. And for example for $PGL_N$ this means that multiset of eigenvalues of $\tilde{s}$ is invariant under multiplication on some $N$-th root of unity. So I am asking for similar classification for other types.