My guess is you have to implement this type of tooling yourself as custom in house tools. I assume game developers / game development companies make their own test tools for this, unfortunately, I assume most (if not all) do not share it with the community as open source.
Then again their tooling logic could be specific to the games they make. Also, then again, if the tools were shared, users can abuse the tools to do DDoS on the game servers, or use it to automate their gaming to gain experience or in game money, etc. to unethically benefit.
Rather what you could get is piecing the individual components for how to automate them and then combine them all together in a test framework yourself.
Game action automation can simply be GUI automation via Sikuli or Eggplant, and other tools, there are more if you look around. Depending the type of game implementation, if they don’t have good GUI control elements that can be controlled via Win32, Windows API, Windows UI automation (e.g. TestStack White, Ranorex), then you’d have to go with visual recognition tools like Sikuli and Eggplant.
Game actions could also be automated at the script and API level if the game offered any SDK/scripting support (python, Lua, etc.) or (REST) APIs to call.
Websockets is simply a special type of HTTP / REST API automation to do. You could individually research websocket automation/scripting. Then you just figure out the game’s specific custom websocket logic/implementation to automate against (the request/response is not the same for each websocket service).