Websocket for real-time quiz application

I am in the implementation of a “one server-many client” type application, which means quiz question should be broadcasted from a privileged user to all the users and after that results should be collected. Question is the web service part, where real-time communication should happen.
There are two types of users in this case, privileged users are, who can send answers to another users. Real-time means in this case, that question have to appear immediately after sending to standard users.

The communication flow should always be user <-> server <-> user.
Everything else is going to end in a burning train wreck.

Communication happens everywhere it needs to happen, it’s just a means to get the data across, what exactly is your question?

I would like to know more about websocket implementation.
Maybe should be Firebase also suitable for this purpose?

Are you constrained to using HTTP? If not, then simply use a TCP/IP connection with a networking library, such as Kryo/KryoNet.
Otherwise you would need to use a whole HTTP server/servlet container. Keep it simple.

^^This

Or, if you want a simple http-wrapper take a look at the LibGDX Net package.
https://libgdx.badlogicgames.com/nightlies/docs/api/com/badlogic/gdx/Net.html