It turns out push events are not triggered when commits are created by github actions via the GitHub API. So let's simplify things and just update the meta branch in the same workflow that updates the casks themselves.
Homebrew tap for binary builds of Emacs provided by jimeh/emacs-builds.
The following Homebrew Casks are provided by this Homebrew Tap:
emacs-app
emacs-app-nightly
emacs-app-good