Let $C_2$ denote the cyclic group of order $2$ and let $C_2 \ltimes U(n)$ denote the semi-direct product of $C_2$ with the unitary group, where $C_2$ acts on $U(n)$ by complex conjugation.
I want to calculate the Weyl group $$ W_{C_2 \ltimes U(n)} C_2 = (N_{C_2 \ltimes U(n)} C_2)/C_2. $$
I believe that the normalizer is $U(n)$, so that the question is equivalent to calculating $$ U(n)/C_2. $$
What is $ W_{C_2 \ltimes U(n)} C_2 = (N_{C_2 \ltimes U(n)} C_2)/C_2? $
My hope is that the answer is $O(n)$, but I cannot prove it. (I do know that $U(n)^{C_2} \cong O(n)$ however.)