LSA instructor pilot describes a student crosswind takeoff that does not go well; resulting in the instructor assuming control and striking a runway light with the tail of the aircraft.

Narrative

Student executed a takeoff [with a] 5 KT direct crosswind from left. Upon application of power; nose of aircraft drifted left; to which a verbal command was given for more 'right rudder.' As the student added right rudder excessively; the aircraft drifted right. I took control of the aircraft as the right wheel was over the right runway white threshold line. As the aircraft was at rotation speed; I rotated the aircraft about 3 KTS above a stall and proceeded in ground effect in an attempt to bring the aircraft back onto the runway centerline. The aircraft struck a runway light just aft of the cabin in the belly. As the aircraft was already at flying speed and insufficient runway remained to recover and stop; I elected to take the aircraft airborne and return via a normal traffic pattern to a landing. After landing and securing the aircraft; a dent was noted aft of the main gear in the center of the belly just at the beginning of the lower taper of the fuselage. The aircraft was secured awaiting a maintenance inspection. Control should have been assumed earlier; but the student had always managed to correct in the past. In this situation the student over corrected; which was not anticipated; and the decision to allow the student to recover was flawed.