It's like I expected, it is a 3/8 and your shaft is 5/16.. Now two more things that are common to putting one of these on. Sometimes the peg is too big to fit inside the fly wheel fins.... or you don't get them centered and they shake your engine apart.
Is your peg steel and hollow?
if so you can weld a 5/16 washer dead center, then slide it onto the shadt and then run a nut and washer inside the peg. Also note that the nut is a 5/16 x 24 fine thread.
Good luck.
Ps I don't think this will help, since i expect the peg you have is an alloy and has one single 3/8 hole all the way through it. But you can find them that are hollow and have only an end cap on one end. With that kind you can do what I describe above.
Measure your flywheel before you go looking, those pesky fins you know. I have one I got off ebay but I'm hanging onto mine in case I ever do this again. Also I can make it work on emotors.