According to my bird books, Gila Woodpeckers (Melanerpes uropygialis) don't like to nest in birdhouses. They usually nest in Saguaros, but I discovered how to get them to use birdhouses (the few Saguaros in my neighborhood look like Swiss cheese and all the holes are taken, so birdhouses are sorely needed). I use a standard woodpecker house with nothing inside but bare wood and mount it sideways under an overhanging roof about 7 feet above the ground.
A baby Gila Woodpecker can be seen below sticking it's head out of a sideways woodpecker birdhouse.
I place a thin strip of wood with a 3/4 inch (1.9 cm) hole in it across the larger entrance hole. Gila Woodpeckers like to excavate their own holes, and because the 3/4 inch (1.9 cm) hole is too small, they will happily enlarge it to the proper size themselves. This fools them into thinking that they made the house themselves. You have to put a hole in the wood strip covering the hole so the Gila Woodpeckers know where to start chipping away the wood, otherwise they will tap on the birdhouse, hear that it sounds hollow, and then begin making a hole, often on the wrong side of the birdhouse such as the bottom. Gila Woodpeckers love to excavate the bottom side of birdhouses for some reason, but it is not a good place for an entrance hole because the eggs and babies would just fall out.
Gila Woodpeckers are fascinated by any small hole into a larger wooden cavity, and so the hole in the wood strip will quickly attract a curious Gila Woodpecker. The woodpecker will look with one eye into the hole, then look with the other eye, and then compulsively chip away at the hole until it is finally large enough for it to look inside with both eyes and solve the mystery of what's inside. If Gila Woodpeckers are damaging exterior wood on your home, putting up a birdhouse or two for them will help divert them from their more ill-conceived and costly excavations.