See also the official documentation:
Help Center > Group Owners Manual > Inviting people to join a group

Owners and moderators who have been given permission (See Members, Role, Moderator Permissions) can invite people to join your group as members using the potential member's email address. 

This works with all groups, including basic (free) groups. Normally though, you would use "direct add" rather than invite with Premium or Enterprise groups.

• Go to Admin, Invite.
  
  
• You can customize parts of the message to improve acceptance rate. (See examples below.)
  
  
• You can include a notice, if you wish. For the type Invite, you can set up as many notices as you'd like, or none.  If you have at least one, it will be in the Notices box on the Invite page in the Customize Message area and you can choose to insert it into the standard invitation.  If you have several, you can choose which one of them to use.
  
  
• You  can also create a Welcome notice that includes your terms of service or other info you wish new members to read. See the wiki on Notices.
  
  
• Type in the email addresses of people you want to invite. Or upload a list. But it needs to be in plain text.  It can't be UTF-16LE.  It must be UTF-8 (it works both with or without faux byte order mark).

See the page about the 20 member limit of invitations in a 24 hour period. This applies no matter which plan you are on - free, premium or enterprise.

• Send invitations.
  
  
• You can track who has accepted on the Sent Invitation button on the Invite page.
  
  
• You can Cancel/Remove Invite - this just removes the email address from this list. It doesn't remove their membership in your group if they have accepted. Or you can send again to prod them to reply!

NOTE: People you invite will be automatically added to your group when they accept the invitation, even if you have a restricted group. 

Examples of Sent Invitations

Standard message sent by Groups.io:

Hello,

You have been invited by Invitor to join the Groups.io group GroupName.

If you have questions about this invitation, send them to OwnerAddress.

To accept the invitation, please reply to this email, or you may accept the invitation on the website. This link will expire in 14 days, on Date at Time.

If you are not interested, or if InviteeAddress is not your email address, please ignore this email.

Cheers,

The Groups.io Team

With an Invite notice set up:

Hello,

You have been invited by Invitor to join the Groups.io group GroupName.

The following message was included by Invitor:

_________________________________________________

By accepting this invitation using the information below, your subscription will progress more quickly. If you apply for the group online or by sending an email, there may be more steps involved.

_________________________________________________

If you have questions about this invitation, send them to OwnerAddress.

To accept the invitation, please reply to this email, or you may accept the invitation on the website. This link will expire in 14 days, on Date at Time.

If you are not interested, or if InviteeAddress is not your email address, please ignore this email.

Cheers,

The Groups.io Team

Notes:

In the example with the Notice included, the lines and the information between them is the Notice that was set up.  The Lines can be useful to draw the person's attention to that area.

Invitor will be the Display Name of the person that sent the Invite.

GroupName will be the actual name of your group.  Format can vary depending on whether you have subgroups or not.

OwnerAddress will be the actual owner email address of your group.

InviteeAddress will be the email address of the person invited.