Fox & Fox
(click on pictures to enlarge)
This one is marked only on the cap: "Fox Permagraph Patd. Fox & Fox N.Y.C. 14k Gold Filled."
Other than the barrel construction, which includes the brightly colored bands, there's not anything mechanically unusual about this pencil; it is a typical nose drive piece from the 1920s. I have not yet been able to track down the patent, so research on this continues.