I put in a new bathtub, and had a plumber do the rough-ins. Now that the tub is tiled and everything, I went to install a cover over the tub overflow hole and realized that there is no small threaded hole(s) to screw anything into, and the overflow is threaded on the outside simply attached to the tub by use of a single large 2" diameter nut. Is this some special brand? I can't find a cover that is threaded like this. Since then the plumber who installed the overflow has gone AWOL and left no info or spare parts.
It could be a Kohler or Gerberit drain, among others, but who ever made it, I don't think you will find any extensions to solve your problem. Especially if the drain closure mechanism is in the overflow opening.