Google Groups Settings + Add Members

Go to groups.google.com

You will then see all the groups that you have access to. 

Click on a group and then navigate to "Group Settings" on the left hand side.

Review all the settings to make sure they are as needed. Important settings to note are as follows:

Message moderation determines if messages need to be approved for them to reach the group. When in doubt turn this on. 

To choose who can moderate messages (who will recive the emails when posts are made for approval) look at "who can moderate content". Also check out this other settings as they are important too. When in doubt go with the most restrictive option. 

To add members, navigate to people-members

Then at the top hit "add members". Fill out the form with the users email under the desired role (owner, manager, member), fill out a welcome message and choose what subscription level (what they will be notified about pertaining to the group).