Old piano Stool repair

My aunt gave me this piano stool to redo. The nut is warm badly and cracked in too. There were some wooden wedges on the sides and I removed then. The nut still will not come out. I sanded the piece in order to look for a hole that was putted over to pin the nut in place but did not see one. Does anyone know how they put these together? Hopefully I can manage to add a few pics to help. Thanks.
Jeff

 

You might look to see if there is anything holding it from the bottom. Otherwise, it might just be snugly fit into the hole.

 

Thanks for the quick reply. The bottom of the nut looks closed. At least that what it looks and feels like. I was thinking of setting up something like a hammer wrench. It's square threads but maybe an acme thread would work. I just don't want to damage it any.