The Axe Soldier (Japanese: アクスソルジャーAxe Soldier) is a class that has never made an official appearance in the Fire Emblem series—Fire Emblem: Genealogy of the Holy War, the only time it appeared in a game's coding, had the Axe Soldier class go unused.
The Axe Soldier would have been one of a number of variants of the similarly-unused Soldier class, others of which include the Sword Soldier, Lance Soldier, and Archer, all of which also went unused; as its name implies, the Axe Soldier would have been the axe-wielding variant.
See class growth rate for more details on the mechanics of class growths in each game.
Class change
The class a unit changes into in Genealogy of the Holy War is determined by the unit and not the class; however, no units change into or out of the Axe Soldier class.