A few thoughts:
First, since you have only 6 variables, are you sure you need PCA? If you expect to get 2 components, you haven't gained that much simplicity.
Second, if you decide that you do want to do it, the wisdom of doing it per group vs. all at once depends on both the question you want to ask and the answer you think you will get. That is, if you suspect that the structure will be different in each group, you should do it by group to see if you are right. You might even wind up clustering the groups by their PCA structure (that would be an interesting problem). If the structures are all different, then mixing the groups may result in mush. You risk the "Switzerland problem" - i.e. On average, Switzerland is flat. Throw the mountains into the lakes and it balances out.