You can add new table that will connect group with calendar, but you will need to write code where you will assign calendar to group. Querying this table you can know if calendar is group calendar.
You will need to extend event module to customize adding and deleting. You can find more details how to extend event module in this post
. But I think syncing calendars is not good idea as it is very hard to maintain everything in sync. Better approach would be to merge calendars when showing them to user, you don't have to take care of calendar sync you just need to read which groups user is member of, read theirs calendars, merge all events and display them to user.