What happens to an airplane when hit by lightning?
Since the aircraft is not grounded, the physics are the same as with a car that gets struck by lightning. The aircraft's skin will be charged, but very little current will be induced in internal parts in the process. The outer skin will act as a Faraday cage and protect the inner parts from the electric field.
This is greatly helped by a metal structure. Composite aircraft parts need a copper mesh over all external surfaces to give them sufficient electrical conductivity. The terminals of that mesh must be connected to the metal parts of the structure to make sure that all parts will have the same electrical potential in case of lightning strike.
However, the ionized air caused by lightning will leave burn marks on the surface, and an inspection of the affected part and the electrical equipment is advisable.