See also the official documentation:
Help Center > Group Owners Manual > Managing moderators and additional owners > Making a member an owner or moderator

If you want to promote a member to moderator or owner or make changes in a member's permissions:

• Go Admin, to the Members tab at the left hand side of your group's website.
  
  
• Either by scrolling through the list of members presented, or by searching on their Display Name or email address (or just a part thereof), find the email address you want to change.
  
  
• Click on the "member's line" anywhere at the left hand side of their entry in the list. (Don't use the checkbox. The checkbox has other options, but does not promote someone to owner or moderator.)
  
  
• That "click" will take you to the page detailing that person's Membership. The fourth section on the page is called Role. It will (or should!) say Member. 
  
  
• At the right hand side of the line with "Member" in it is a drop - down arrow.
  
  
• Click on the arrow and a dropdown list will appear with the options of Member / Moderator / Owner available.
  
  
  • Or click on Owner, and "Owner" will appear in the place where Member previously appeared.
    
    
  • Or click on Moderator, and "Moderator" will appear in the place where Member previously appeared.

Decide what status you want the person to have.

Moderator privileges

• Approve Pending Messages
• Edit Archives
• Invite Members
• Create/Delete Hashtags
• Manage Subgroups
• Add or Modify Integrations
• Modify Group Settings
• Manage Paid Subscriptions/Donations
• Billing
• View Member List
• Approve Pending Members (also allows access to the member list)
• Ban Members (also allows access to the member list)
• Set Member Subscription Options (also allows access to the member list)
• Set Moderator Privileges (also allows access to the member list and allows setting member subscription options)
• Remove Members (also allows access to the member list)

Optional notifications

• Email when there are messages needing approval.
• Email when there are members needing approval.
• Email when a member joins/leaves this group.
• Email when group storage limits are reached.
• Email when someone creates or deletes a subgroup.
• Email when someone reports a message.

Change the permissions - Owners have all permissions, while moderators can have specific permissions. 

When you are done, be sure to scroll to the bottom of the page and click Save.