Let $a$ be the following matrix in SU(N)
$$ A=\begin{bmatrix}a & \\ & a\\ &&\ddots \\&&&b\\&&&& b\\&&&&&\ddots\end{bmatrix} $$
with $m$ number of $a$ and $N-m$ number of $b$. What is the space of $g$ such that $B=gAg^{-1}$ is a complex symmetry matrix i.e. $B=WW^T$ for some $W\in SU(N)$?
Here is some thing I know. If $g\in SO(N)$, $B=gAg^{-1}=gAg^T$ is a symmetric matrix. Hence, $SO(N)$ is the subset of the space of $g$.