Amplifying a pulsed current
Quick, terse answer pending answers to my questions above: you don't want to amplify the function generator's current; you want to amplify the function generator's voltage into a current in the coil.  The name of such an amplifier in general is a Transconductance Amplifier or a "Current Amplifier".

Unfortunately, searches on those terms will get you a lot of examples that are too wimpy for what you want.