We've just made a fresh install of Joomla 3.6, AVChat 3.7.2 and the Joomla! Component (build 902) and groups and permissions worked as expected.
What permission are you unable to give to what user group?
Each Joomla!'s user group permissions are influenced by the permissions of it's parent user group. Here is Joomla!'s standard groups tree with parent and childs:
For each permission you have 3 options:
* Inherited (default)