The auxiliary version of the Icy Overkill core only has a cooldown of 2 turns despite that the normal auxiliary cooldown is 3 turns (unless this was intended as an extra effect?). I would imagine that this was due to copy pasting of the original sidearm version of the core in which a 2 turn cooldown aligns with the normal 2 turn cooldown for sidearm.
Cooldown. If you use aux, 2 turns later you can use Icy Overkill even though the regular aux is still on cooldown for one more turn like it's supposed to.