Place the head of the axe into a table clamp with the shaft of the axe pointing up at about a 25-degree angle from the table. Tightly clamp the axe head in place and put on a pair of leather gloves or gloves of comparable thickness and protection. The gloves protect your hands from splinters, cuts and bruises later on.
Saw off the axe shaft as close to the head of the axe as possible using a coping saw. After sawing off the shaft, check the axe head for any steel wedges or screws inserted in the piece of shaft still stuck in the eye of the axe head. Remove any wedges or screws, using a drill.
Set up two axe blocks. These are cubes or three-dimensional rectangles of wood with grooves down the center on top which support an axe head while it is worked on. Alternatively, clamp the axe head in place upside down so there is room below the eye.
Drill out the center handle material in the axe eye, working from the underside of the axe which should face up. Not all of the material will drill out, just work on removing the core of the axe shaft.
Set a wooden or steel drift on top of the remaining shaft material in the axe head while the axe is still clamped upside down with space beneath it. If applicable, choose a drift specific to the type of axe you are working on, as some drifts come in varieties for double-bit or single-bit axes.
Hit the drift with a mallet to push the handle material out the top of the axe head. The top side of axe heads have a wider opening for the insertion of wedges, so the handle material exits the eye of the axe by hitting it out from the bottom.