I needed a short extension for my 3⁄8-in.-drive Forstner bit. I went to three hardware stores and came up empty. I did, however, find the materials I needed to make an extension from scratch.
As shown in the drawing above, I bought a length of 3⁄8-in. by 16 threaded rod, and a 3⁄8-in. by 16 coupling. Then I got out my 3⁄8-in. by 16 die (if you haven’t got one, they cost about $4), and cut about 1 in. of threads on the bit’s shaft. This worked just fine on my inexpensive Forstner bit, which has hardened-steel cutting edges and a much softer steel shaft.