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Well, first of all, a solenoid is a tightly wrapped helix of electrical wire, wrapped around a ferromagnetic core. When a current is allowed to flow through the electrical wire, a magnetic field is produced in the inner core. This is the principle behind all electromagnets. If you remove the ferromagnetic core, it loses its electromagnetic properties. The current flowing through the wire helix is still the same. If you couple this solenoid with another, to produce a transformer, and then try removing the core, you would get an equalizing effect, as a transformer is designed to either step-up voltage or step-down voltage. If you remove the primary core, there is no magnetically induced field to pass along to the secondary coil, so the current would remain the same. The same would occur if you remove the core of the secondary coil, there is no receiving core, so the current passed through the primary coil would be the same.
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