For a given fixed matrix $M\in SU_{\mathbb{C}}(n)$, how to find all $N\in SO_{\mathbb{C}}(n)$ such that $N^{-1}MN$ is a diagonal matrix?

If we consider a fixed set of $n$ complex vectors $\Gamma:=\{x_1\cdots x_n,x_i\in\mathbb{C}^n\}$, then the above problem is equivalent to finding all matrices $M\in SU_{\mathbb{C}}(n)$ such that it has $\Gamma$ as its eigenvectors(with possible different eigenvalues.)

One step further, what will the answer change if $M\in U_{\mathbb{C}}(n)$?

For a given fixed matrix $M\in SU_{\mathbb{C}}(n)$, how to find all $N\in SO_{\mathbb{C}}(n)$ such that $N^{-1}MN$ is a diagonal matrix?

If we consider a fixed set of $n$ complex vectors $\Gamma:=\{x_1\cdots x_n,x_i\in\mathbb{C}^n\}$, then the above problem is equivalent to finding all matrices $N\in SO_{\mathbb{C}}(n)$ such that it has $\Gamma$ as its eigenvectors(with possible different eigenvalues.) So the totality of such $N$ consists of a subgroup of SO_{\mathbb{C}}(n)$$

One step further, what will the answer change if $N\in O_{\mathbb{C}}(n)$?