The iPhone (iOS) allows for very little control over contact groups. There is no way to create a group on iOS but there is a way to create a contact in an existing group. This is done by tapping the "Groups" button in the upper right corner. There will likely be several options here for groups. Tap the one you want to create your contact in, and you should see checkmarks on the other groups turn off. Then tap the "Done" button. You can now tap the "+" button to create a new contact which will be added to the group that you are in.
Note: the following instructions work if you use iCloud for managing your contacts.
To edit groups you must either use iCloud.com, or have a Mac and then use Contacts.app. So, you can accomplish your stated goal by:
- Going to iCloud.com, logging in, and selecting "Contacts".
- Select the red group icon towards the right of the left page.
- Click the "+" button on the bottom right of the left page.
- Enter the group name.
- This new group should appear on your device.
- Tap the "Groups" button in the upper right corner of Contacts.app on your iPhone.
- Select the group you just created.
- Tap done.
- Create your contact.
Someone can then log into iCloud.com and change the group that the contact is associated with.