External notification

Hmmm, the latest update is not logging any Activity so there's notification happening. The group Activity post is there but it's not logged as an Activity event or as a Notification.

It's not logging any activity for doing what exactly? Notifications don't send on activity logging. They send on notification logging. Specifically those configured under CB Activity > Parameters > Notifications. Which of those isn't working for you?

If you were using the old CB Activity > Parameters > Notifications > Activity > Create (Groups) notification though it's now gone in 7.0.0. It is instead entirely integrated into CB GroupJive and will respect whatever configuration you've set within CB GroupJive. The same applies to CB Activity > Parameters > Notifications > CB Gallery > Create (Groups). They're both subscribed to via the GJ notifications page and will send as a GJ notification.

So I'm not using the GJ Wall, I'm using GJ Activity instead, so I have people posting to specific Groups via GJ Activity, then there's an Activity page in the various groups and general Activity wall that includes articles, GJ events, GJ Gallery, etc, for all groups a user is part of.
That's all posting fine. But before this update it also used to post to the internal notifications (CB Activity > Activity) and that would trigger a Notification (the little blue mail icon and globe icon on the user's profile page).
And it would trigger the Auto Action I had setup to send a mail. 
Since the latest update that trigger doesn't happen (CB Activity > Activity), so no mails are sent and no internal notification is logged.

You don't need to be using GJ Wall. The "CB Activity > Parameters > Notifications > Activity > Create (Groups)" notification works the same as all GJ notifications now. As in you need to navigate to your group, go to its notification page, and toggle on "Create of new activity post" notification.

This will send a notification based off whatever you've CB GroupJive configured to do same as other GJ notifications. So probably an email by default. If you also want it to log a CB Activity notification then navigate to CB Activity > Parameters > Notifications > CB GroupJive and set "Notifications" to "Enabled" or "Replace". This will allow GJ notifications to also be logged into CB Activity.

I know, I said I don't use GJ Wall.
OK, thanks... CB Activity > Parameters had been switched off in the latest update, they were on before.
But my older system is back on now, thanks.

On the email setting in CBActivity > Parameters > Notifications > Email, it seems to send a notification to everyone in the database, not to those in the GJ group only.
I'm hoping for something that is very specific to GJ, so posts to everyone in the GJ group, comments and replies to those in the discussion.

To send a notification to a group when a post is made on a group your users need to subscribe to the "Create of new activity post" notification notification in that group. This works the same as all the GJ notifications. If it's sending to everyone then you turned on the generic notification that isn't group specific so that's why it sends to everyone.

Group Notification Menu
 

Group Notifications
 

That is how they can subscribe to that notification now. This works exactly like all the other GJ notifications on that page. This is no longer in CB Activity > Parameters > Notifications. The reason is that was a bandaid implementation that ignored and bypassed normal GJ notification behavior. That is now fixed and works the same as all the other GJ notifications.