Corps of Engineers VS Service Commands of WWII

Can anyone elaborate on the how support commands were structured? I.E. I have done research on the 7th Air Force and found that it consisted of the 7th Fighter Command, 7th Bomber Command and the 7th Air Force Service Command. How do the Air Force Service Commands tie to the Corps of Engineers as the AFSCs were in charge of support/maintenance etc?

Example: 804th Aviation Engineers Battalion were ones charged for building the airfields on Makin & Saipan (which is where the 7th operated from). Were they part of the 7th Air Force Service Command? Or were they part of the Army Corps of Engineers? Or Both? Im having trouble figuring out the structure.