I have looked in my Ackley books Vol. 1 and 2, cannot find any reference to a Ackley improved 22 Hornet.
Chances are it is a 22 Hornet rechambered to 22K Hornet. The only way to find out what you have is to compare fired case's or a chamber cast.
I think the seller is trying to make the rifle sound better than a 22K and is using the Ackley name.