Today I am going to talk about this SharePoint behavior which is by design and does seem to confuse some users.
When you create groups under Peoples Groups you have an option to specify Group Owner. Now you would think making someone a Group Owner would allow them to add users to that group.
But this is not how it works. For any group owners to be able to add users in the group they need to have Manage Permission permission level assigned to them.
To create custom permission level and assign that to SharePoint group you can follow the steps mentioned below.
How to create and manage permissions level