First off prepare some wedges.
I’ve used Oak and split out some sections from a larger piece that’s been sitting around for a while so it’s nice and dry. Put them in a low oven at home for an hour if you want to make sure. The pictures give you a guide to the shape and size of the wedge, each is slightly different depending on the shape of the heads eye.
I split the Oak down using the growth rings running the length of them and shaped the end so as to not be too weak, basically add a v shape secondary bevel. Don’t be fooled by the other short numerous markings in the oak, they are the Medullary rays which run at 90 degrees to the growth rings and when in the round stem, add Oak it’s strength.
Start to look at the head on the (now dried) helve and check everything is the right way round then pop it on the top and draw out the eye shape onto the top of the wood. Start shaping with axe and knife until you can start to offer up the head. Tap it down a few times then take off to see the areas that need to be further removed. Scraped areas and dark marks show you where.
Once the head is on or before if you like, split the wood the length of and centrally along the carved out section with your knife, using a baton. Then fit the Oak wedge also with the baton, hitting it down until well in. You can put a spot of wood glue on to make sure of a good fit if you like. The wedge due to its length can then be used a second or third time for other heads, after a reshape with the Axe.
All done, with a slight bend in the handle and head for right hand use, but you could make it centrally located for general splitting and chopping work or to the left. The choice is yours.
Finish with a drink of Raw Linseed Oil (or other suitable oil) around the head over night and a wipe over the rest of the helve, then allow curing for a couple of days.
A quick wipe over with an oiled cloth now and then and you should have a nice old tool perform well for many more years, and saved some money too.