As of iOS 16, the standard iPhone Contacts app allows you to edit your groups. Apple has renamed groups to lists.
You can also edit your groups 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, select Groups, then select the desired group, and then 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:
Need help? You can email us at firstname.lastname@example.org