We are struggling though with getting the following variables out of the system: Group name, group creator id, group participants ids
You'd need to specify what trigger you're using.
I know that you have already a variable that outputs all the participants of the group (since you need them for the notifications that you have already biult).
What is that variable?
There is no variable for that. A trigger is planned to be added to GJs sendNotification function soon (after done with new CB Activity notification API) to allow overriding the notification behavior, which will include the group object, group user object, and cb user object. It'll likely contain more information as it needs to also have for example the event object if the notification is from GJ Events. This will be used in System Actions to log CB Activity notifications instead of send emails, etc.. for example.