How Does Group Owner Add Members Without Full Control?

We have a site for one of the departments with a group set up with Design and Contribute permissions. We would like for members of that group to be able to add members both to the same group, and another group with Read permission. I have not been able to configure a group to be able to add users unless they have full control, even though I have added the permission as shown here:

Am I trying to get a square peg into a round hole?

Echo Train Nashville TN