What exactly are you trying to do? Do you actually need a HomeGroup set up for something specifically?

If it's set to Public, there isn't really a problem unless you are trying to do something and it isn't functioning. Public is the most secure mode, but yes, it doesn't allow Homegroups.

You should be able to change from Public to the more open Home or Work option by just going to the network and sharing center and clicking on "Public Network." Does it not allow you to do this?