By default users will receive a notification based on their preference, when some one invites them to a group.

I have all sites notifications on and when an invitation is sent it's not being sent to the site messages.  I do notice that under group notifications it defaults to both site and email notifications but only after the user has joined.

By default the group admin has got full control all on contents inside a group.

@Team Webgalli 

Understood, upon doing further testing this stands true for all content EXCEPT pages within groups... any idea why?

I think the default access level pre-selected on new content is not necessarily "private" but whatever you selected as default access level in the advanced site settings. There is very likely no difference between group and non-group content regarding what you have set as default access level and what is pre-selected on new content.

Pre-selecting the specific group as access level is possible somehow for sure. But I'm afraid it would require some customization for sure - and I can't tell right now what exactly.

On Elgg 1.9 the behaviour will be partly different:

• for open, unrestricted groups the pre-selected access level will still be the same (i.e. site-default access level),
• but if you make a group a restricted group (= non-members should not see the content) the only available access levels for group content will be "group" and "private". "Group" is the pre-selected access level in this case.

It's still not a perfect solution in any case (e.g. existing content in restricted group will keep their old access level) but it's at least a progress compared to Elgg 1.8.

There isn't a way included by default, but it's easy functionality to add via plugin if you're a developer.  It's a matter of copying the value of the history annotation to the page entity description and saving the page entity.  Probably would want to create a new history item with the new value as well.