When User Trying to Modify Member list in Outlook they will get the below error Message..
"Changes to the public group membership cannot be saved. You do not have sufficient permission to perform this operation on this group"
1. Check the distribution group/Mail Enabled Security group Membership Tab in Active Directory and Make sure correct user name were assigned and "Manager can update membership list" were checked.
Now you added the desired user list in the required group Membership list.