It is marked Craftsman with an F in a circle and Made In USA.  About 18 inches long, the barrel end about 11/16.  The knurled protion will turn but has limited movement - rust for some other physical restriction to turning.  A tab moves thru an L shaped slot - but again it actually doesn't move.  I don't want to force these things without knowing what it is supposed to be doing.
The barrel end has a square opening just a fraction over 3/8 inch and inside is another square which is not aligned with the outer square.  The L shaped slot has a bent spot - it looks like something was forced at one time.
Here's photos:

Bit extension.  Soak the working end with some penetrating solvent & gently push the drive end of an auger bit into it.  It should lock in place.

It's an extension for those times that you need to drill really deep holes, or drill holes in hard-to-reach places, like through the fireblocking in a wall.  It works with square taper shanks, like the square taper on the end of it.  I'd have to have it in my hands to figure it out, but it probably starts with the two square openings aligned so you can insert the bit, followed by the squares getting out of alignment to trap the bit, and the inner square traveling somehow to tighten the bit up against the end square.

Start with lubricating it, then fool around to see if it becomes clear how it works.  There were multiple different "chuck" designs on these extensions.

I see Geneg types faster than I do, but maybe the two of us together have provided useful information.

Thanks, guys!  I will get it soaking tonight.

I did type faster, but had a typo.  It, not, I should lock in place.   I've found that letting things like this soak in a small can of atf or Marvel mystery oil for a few days loosens them up with minimal effort

Hey Al. I might have one of those here, I will take a look. The one I have might be a Millers Falls, my guess is that would have been who made it for Craftsman.

From the craftsman code list:

F-circle = possibly Lectrolite, Miller Falls, or KD, ca. 1949 - 1964

Hello, Al. Turns out I don't have a Millers Falls here. I am pretty sure though, that yours is the same as a Millers Falls No. 11  Bit Extension. From looking at the catalog photo, that bump in the collar is correct.Regards, Lou

The bump is supposed to be there?  I thought it was damaged.  I haven't taken it out of the solvent soak yet - later today we'll know if it has freed up.

The bump is there so that the sleeve can rotate to hold the bit in.  Similar to some of the screw holding screwdrivers.