What is the term for a solenoid that has an iron core inside it?

A solenoid with an iron core is referred to as an electromagnet. The presence of the iron core enhances the magnetic field produced by the solenoid when an electric current flows through it. The iron core becomes magnetized, greatly increasing the strength of the magnetic field compared to a solenoid without a core. This property is utilized in various applications, such as in electric motors, relays, and magnetic locking systems.

By definition, a permanent magnet is a material that maintains a persistent magnetic field without the need for an external electric current, while an induced magnet refers to a material that temporarily becomes magnetized in the presence of an external magnetic field but loses its magnetism once the field is removed. A simple magnet is more generic and doesn't specifically involve a solenoid or an iron core. Therefore, the term "electromagnet" is the most accurate description of a solenoid with an iron core inside it.