As Arthur Evans said:
I think there are two competing schools of though here (at least):
1) We should expose a title on the tab group.
2) The action bar title should take the title of the current tab's window.
I've spent some time reviewing the Android docs, and it's not clear to me which of these is preferable.
IMHO, if the tab already has a title, it's quite pointless to repeat this title in the ActionBar. In my application I'm using tabs to split the content of a shopping list between the items and the values... one tab is called "products", the other is "summary", and the actionBar title would be the list's name. I've attached a screenshot as an example.
Bottom line: Maybe the TabGroup could use the provided title property, and when it's not present, use the active tab title?