The official stance on the three default groups in Google Contacts (Family, Friends and Coworkers) is that they cannot be renamed nor can they be removed.
Now I am not going to argue this point, mostly because, unlike Google Alphabet, I am not worth some half a trillion dollars.
What I will say is that, if you have been using anyone's service long enough, you will find it possible to be an edge case, where the regular rules don't apply, and suddenly you have figured out how to get around the official stance, and naturally, no one else has come across the exact some problem. Luckily, there is a way to solve this particular issue. At least, until they close the loophole.
Unfortunately, I cannot tell you how the problem came about, though I am sure that it was my own doing. Most likely, it was when Google rolled out the default groups a few years ago. I don't like it when anyone sticks their defaults down my throat, and so when I saw new groups in my contacts, I deleted and/or renamed them. At the time, it was probably possible to do so.
Later, when other services (for instance, Voice), became integrated with the default groups, it was determined that you should not do so, and the window to do so was closed. The problem is that now I had removed the groups, and so I cannot actually recreate them either. It's even stranger, because I have the default groups in my Google Voice account, but I can't see them in Google Contacts. Uh-oh.
Anyway - how to fix the problem.
If you look at the link for one of the default groups, it looks like this:
Fairly straightforward, right? Note that your own link may be a bit different, specifically, if you use multiple accounts, the "0" may be 1, 2, 3 or something else entirely - but the "family" will not change (unless, of course, we are talking about "friends" or "coworkers"). So go ahead and click the link - you should end up in your own contacts. If you see mine, however, please let me know.
Meanwhile, if you look at another group, that is one that you have created yourself, that link will look much the same, but instead of "family", "friends" or "coworkers", you will have a group id, which is an alpha-numeric id of the group in question. Well enough, but it doesn't solve the problem, because the id is created for you, so creating new ids is where we need help, how does that happen?
That's where we need to trick the system.
By using an application that already has those ids built into it, we can recreate those groups automatically. In this case, the Android Contact Manager. Now you don't have to use your actual phone (though you can if you like). Many old phones will work, but they do need to have the group function in the Contact Manager App. This means a really old phone (say, with Froyo (Android 2.2) won't work - no groups. However, I tested Android ICS (4.0.4) and it worked just fine. Others might as well, so your own mileage may vary.
Here's what you need to do:
First, you'll want to connect the phone (a tablet may work as well) to your existing contacts, assuming you want to keep them.
Next, verify the counts. Depending on the device, you can typically view by group to see the number in each group, and compare by checking the group counts in a web browser too. If you're off, then it might take a re-synchronization to take care of the problem, but you'll want to make sure both sides are good before continuing.
Once you're all set, you should see that whichever group(s) you need have a count of 0. Say you need to add "Family". Simply choose a test contact from one of your other groups, and change it to be in the Family group. When you sync the change back to Google, you will see that the Family group has been recreated for you.
You can repeat this as needed for the other missing groups (Friends and Coworkers), and once you are done, you can use the same method to change/move records individually, or you can use the web interface to move multiple records between groups, and then remove existing groups if you would like. Also don't forget to check your Voice settings if you have changed things around!