math syntax highlighting with auctex in emacs

Question

I am using emacs and the package auctex to write my latex documents. I would like to improve syntax highlighting (or font locking in emacs' jargon) for mathematics. For example, at the moment everything in between two dollar signs is the same color, except for ^ and _. I would like to add the following things:

• have a separate color for $ (I don't use $$ at all);
• have a separate color for things like \times or \alpha inside math environments like $...$, \[...\],\begin{equation}...\end{equation} and so on.

For instance, an expression like $G\cong H$ should show three different colors. One for $, one for G and H, and one for \cong.

I have looked for answers to these questions but haven't found any good solution, for example here the OP is asking essentially this same question, but no proper answer.