Automatically filling field in related content type
So if I undersand you don't have any relation Groups -> Projects? You just have a relation Projects -> Groups (that you also call parents)? .
And you want to show by views, Groups and their children (projects)?

If this is what you want you have several choices.


Add many relationships in the view advanced filters in order to reach groups projects, (could be tricky).
Add a field entity relationship Groups -> project that you programmatically set on every projects insert/update/delete. Thanks to this it would be much more easier to build a view.
create your own custom block, with your dedicated SQL query. listing all children of a group and attaching this block to all "groups" content type pages.