Can we give badges on grouping and group wise basis.?
Not directly, but yes combining activity restrictions with completion: you can create an activity visible only for users in Group/Grouping X, and completion criteria with something link "Student must view this activity to complete it". Then, if you create a badge with the "Activity completion" criteria based on that activity, only users belonging to a group will be able to award it (although they will need to access to the activity to earn it). It's a little bit tricky, but it should work