Finished my axe and really - its more like a tomahawk.
I started with a piece of mild steel stock 1/2 inches thick, one inch wide and four inches long.
This was heated and drifted where I wanted to make the hole for the handle. Interesting for me was the shape left behind after drifting as you go from both the top and bottom of the piece - the hole in the steel forms an hour glass shape. This means the handle is compressed about the center in the hole in the steel and after you drive your wedge, the wood is driven into the top of the hour glass and is forced into place. Very difficult to dislodge and modern heads of hammers and axes are machined so this function is no longer done in manufacturing.
As I drifted, one side of the hole expanded more than the other - it was not evenly heated so I had to cool the bulging side in water, which causes the steel to stay put while I drove the drift and the hot side eventually bulged to match.
The bit is made out of a farrier rasp. To weld it into place I split the mild steel on the end and forged the piece of rasp into a wedge and fit it into the split.
Forge welding is done at a temperature that is close to the point where the steel starts burning, bright yellow but not white hot which is when the steel actually combusts and sparks off.
The problem with steel is that under temperature it will create oxide immediately when it is taken from the fire. You can't weld if there's an oxide barrier between the pieces of steel you wish to weld. I used borax as flux which melted into the cracks between the mild steel and the higher carbon farrier rasp. Once melted, I got the bit hot enough to weld the steel together. Its a gentle tap tap tap to weld, not the big sparks you see in movies. The tapping compresses the metal just enough and creates a momentary molten condition as the heat is concentrated during the compression. Magic, both pieces of steel welded together.
The tomahawk is now 5.5 inches long with a 2 inch blade and carbon steel extends back into the handle about 1.5 inches. Essentially its laminated in place - the center is tool steel and the outside is mild steel
Handle is ash treated with linseed.
Bit was hardened and heat treated