Adding a supergroup to InviteMember is very similar to adding a channel.
Click Telegram Group:
Promote @InviteMemberBot to supergroup admins. It only requires one permission — Restrict Users (or Ban Users):
You might also need to change Who can add members setting to Only admins:
Now you can post a message in the group and mention @InviteMemberBot.
Now go back to @InviteMember, you will see a confirmation: