fixed effects model: controlling for different groups

Hello.

I have got panel data and use fixed effects model for my regression.

Now I want to control for different groups. As it is not possible to include a dummy variable for the group (because it is time invariant), I just seperate my regression and make one regression for each group.
When I get my results, can I just compare the outcome or do I need to make some tests to compare the mean results?
Can I Interpret for expample that one variable is significant in one regression and not in the otherone, or that my R^2 within is higher in the one reression than in the other one?

I am looking forward to get a response.

Thanks in advance
Lisa