Precision made in Sheffield, this is a superior component which actually has induction hardened steel blades unlike most of the others made from soft steel. It has many uses.
It is used to enlarge ("counterbore") the 5/16" lamp auger hole to 1 inch diameter to a depth of around 1,1/2". This is necessary if you want to make a standard lamp out of two or more sections. You turn a 1" tenon on the end of one section so it fits and glues into the "counterbore" in the next section, accurately. The tool ensures that the auger holes stay in line so you can feed the electric flex right through the finished lamp.
The tool is also used in lamp drilling to drive the wood. You start with a normal drive centre and drill (through the tailstock and guide bush) part way through the length of the wood. You have to stop somewhere or you will damage the auger by hitting the drive centre if you break through. You then have to reverse the wood to drill the other end to complete the hole. This is where the counterbore/drive comes in. The 5/16" pilot fits into the auger hole and drives the wood, replacing the normal drive centre.