Although the standard iPhone Contacts app doesn't allow you to edit grousp, you can certainly do so with the Contacts Sync app. To edit your contacts and groups, open the app and navigate to the "Contacts" tab.
To add a contact to a group, go to the "Contacts" tab, select Groups, select the desired group, and press the "+" button.
For a more complete contact management solution, you may want to consider "A2Z Contacts." You can download it at: https://apps.apple.com/us/app/a2z-contacts/id622492105
This help article relates to the iPhone app called "Contacts Sync" which can be downloaded here:
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