iPod touch 4th gen is able to run IOS 7, But...
The iPod touch can start on an IPod touch 4th gen, Multitasking works, control center too, you can watch videos and listen to music, but the camera doesn’t work at all, and you can’t connect to WiFi or to pair bluetooth devices, and you’ll see that the software runs very slow due to the ram limitations,also installing the software via iTunes is very complicated and should be done by someone who knows what’s doing, and you’l have To keep in mind that the setup.app file on iOS 7 does not work on an iPod 4th gen so you will have to delete it before installing the software,but installing this Os Is completely useless because an iPod without WiFi or Bluetooth pairing and even without camera is something useless and in my opinion you better stay with iOS 6.1.6 that has Camera, Wifi and Bluetooth and runs Ok, not a software that gives you nothing at all.
iPod Touch (4th Gen)