"Retract" is not an ideal verb to use here. Anglerfish (at least those of family Antenneriidae) have a "rod" (ilicium - formed from the first spine of the dorsal fin) with a "lure" (esca) on the end of it. The esca is waved about by moving the ilicium (with different species appearing to utilise different techniques). The main articulation of the lure is as a hinge, that is it swings forwards in front of the head, and can be placed back so that it sits along the skin.
I seem to recall that a few species have dermal flaps that cover the lure, but these have not been present on the specimens I have kept. There are quite a few species of frogfish that have tiny lures to the point that they are barely noticable, and I am fairly sure that some are dexcribed as lacking a lure.
The article is a bit misleading as all frogfish prefer to walk than swim and many of them cram into crevices, or siddle up next to corals and sponges before changing their colour to match their new home. I may drop Ted Pietsch an email to see if I can get more information on this fish.