Let $X \in \textrm{GL}_n(\mathbb{C})$ be a hermitian matrix ($\space ^t \overline{X} = X$). For another hermitian matrix $Y$, let's say that $X \sim Y$ if there exists a $g \in \textrm{GL}_n(\mathbb{C})$ such that $gX \space ^t \overline{g} = Y$.
I was wondering if there is anything in the literature about modifying the equivalence relation $\sim$ to restrict the $g$ to certain subgroups of $\textrm{GL}_n(\mathbb{C})$. Specifically, I want to know if a complete set of representatives for the equivalence classes are known if we restrict $g$ to be an upper triangular unipotent matrix.
More generally, I'm interested in the following problem: let $\sigma$ be an automorphism of a field $E$ with $\sigma^2 = 1_E$. Give a complete set of representatives for the equivalence classes of $\{ X \in \textrm{GL}_n(E) : \space ^t \overline{X} = X \}$ under the relation: $X \sim Y$ if and only if there exists an upper triangular unipotent matrix $g \in \textrm{GL}_n(E)$ such that $gX \space ^t \overline{g} = Y$.
If the case $E = \mathbb{C}$ has been done before, understanding that may be a good start for an investigation into arbitrary fields.
Special case $\textrm{GL}_2(\mathbb{C})$: Let $X = \begin{pmatrix} x & y \\ \overline{y} & w \end{pmatrix}$ be a Hermitian matrix. So $x, w \in \mathbb{R}$ and $xw \neq |y|^2$. For $a \in \mathbb{C}$, we have
$$\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x & y \\ \overline{y} & w \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \overline{a} & 1 \end{pmatrix} = \begin{pmatrix} x + 2 \textrm{Re}(ay) + w |a|^2 & y + aw \\ \overline{y + aw} & w \end{pmatrix}$$