The shower head stem pipe is rusted through and has broken off inside the fitting pipe that is sticking out of the wall. There is about an inch of fitting pipe projecting from the hole in the tile. I need to figure out how to remove the stem pipe from the fitting pipe without damaging the threads in the fitting pipe. Thanks for your help.

Take skinny flat screw driver and hammer, insert screw driver between the female 90 and broken pipe and gently keep tapping until it gets loose. Don't damage threads inside the 90. The section of shower arm that is threaded is a very thin piece of metal so it is relatively easy to break the grip and remove it.

Do as Milo Dolezal says if that will not work,then You will probably need a Nipple Extrator they sell them at home centers and hardwares. Make sure you get the one for 1/2" pipe see the image below. Good luck.

John