Interaction of a permanent magnet's field with an induced magnet

When you place a material capable of being magnetised into another magnetic field, it will become an induced magnet. (For example placing a nail in the vicinity of a bar magnet).

Magnetic field lines cannot overlap (to my knowledge – correct me if I’m wrong!), so my question is:

How do the fields of the inducer and the induce-ee interact?

Does the parent field bend around the induced one? (Like in my attached image)
… or is the induced field sort of… combined into the parent one?

If the parent field bent around the induced one, surely that would mean that the object (here it is a nail) would no longer be in another magnetic field, and its domains would revert, losing its magnetism?