Pencil protector/extenders used to be available in endless styles, shapes, and sizes. This one was made by the Eberhard Faber Company in the early 1920s. It was nickel-plated, came with an eraser, and had a clip for your pocket.
It was similar to this one from a 1927 stationer’s catalog, though it didn’t come with a clip:
It may be from the line of “Protoclip” accessories:
Complete with Diamond Star logo:
Did the protector/extender lose favor because pencils became cheaper and therefore were looked after less carefully? Perhaps the restriction on materials during the Second World War, which interrupted the manufacture of metal ferrules, heralded a change in thinking.