So, we've ran into a couple folks who have annoying 3.5" octagon or pancake boxes firmly embedded into exterior walls, and want to mount a receptacle in them -- the pancake box from a linked thread is depicted below, along with a dimensional drawing of same.
Unfortunately, the screws provided on a 3.5" octagon or pancake box for device mounting don't fit much of anything besides a light fixture as they are spaced 2.75" apart -- single and duplex receptacle yokes are too big with their 3.28" standard mounting screw spacing, and flanged outlets are too small as their mounting holes are 2" apart, as can be seen below.
So, can you use a raised 4" round to 3.5" round adapter mudring, such as a Crouse-Hinds TP332 or the Raco 727 depicted below, mounted with its 3.5" side to the existing box, to provide 4" standard mounting holes which can then be used for a 4" weatherproof faceplate adapter (Taymac PRBA400G, depicted below with a different box) or weatherproof round extension box as well as a faceplate adapter (Bell 5363-0 or equivalent extension box)? (Of course, you'd have to do some extra work to make sure things are sealed appropriately for outdoor use due to the mudring & box sticking out, but that's beyond the scope of this question.)