No. An induced magnetosphere (like Venus, Mars, Europa, Titan, comets, etc. have) doesn't require or have anything to with the core. It just requires the presence of some kind of atmosphere, in which the magnetic field is to be induced.

An intrinsic magnetosphere (like the Sun, Ganymede, Earth, and the other five planets have) is by definition generated in the interior of a planet, and for rocky/terrestrial planets lile Earth and Mercury this would tend to be in the metallic core (as opposed to the rocky mantle). But gas giants and ice giants generate their intrinsic magnetic fields above their core. For example, Jupiter's and Saturn's magnetic fields are generated in their liquid metallic hydrogen mantles.

An "active core" isn't really a scientific term, and can have different meanings in popular discourse. The usual, better meaning is that there is an active dynamo in the core, generating an intrinsic planetary magnetic field. But the absence of an intrinsic magnetic field and the core therefore not being "active" in this way does not imply the core is solid (let alone not rotating; all cores rotate along with the rest of the planet). There needs to be additional forcing to generate a dynamo. (For example in the case of Earth's core, the freezing out of the inner core causes the outer core to convect. Planetary rotation twists this vertical convective motion into spirals and this combined motion drives the dynamo.)

Often, "active core" is instead or additionally taken to indicate or be synonymous with active volcanism or tectonics. But these are driven by processes in the mantle and crust, and not directly related to the core, let alone the magnetic field. So this idea of an "active core" is "not even wrong".