Now lightly tap it in place to registers its position. Often the punch will stay put when you remove your hand – if not it is easy to relocate.
Now, using a bit that is 3/64″ under the size of the square hole (3/8″ square hole – use 21/64″ bit), drill out the center. Use an ordinary twist bit for this and not a brad point. The leading sharp corners of the brad bit will catch on the inside of the chisel and damage themselves.