At the moment, this is not a feature that we have in the system. We want to make sure that your boards stay as simple and clean as possible. Having the ability to have different column labels for the different groups would make things more confusing.
We have come up with a solution that users absolutely love though! What you can do is create a folder and move everything up a level. The original board should become the folder and each group will turn into a board. This will keep you organized, focused and allow you to customize everything how you want.