What happened between S.Ops and Halo 5?

Hey guys,

I’m confused. At the end of Halo 4 Spartan Ops Halsey was all like: “I’m done with the UNSC, rawr, they tried to kill me. Now I work with Jul.” And Palmer was all like: “I had to kill Doctor Halsey before but now I keep her actively alive!”

And in Halo 5… Suddenly Jul is shot down, Halsey is on UNSC side again, and all seems well. Did I miss something?..