Other mistake: In the private conversation scene with Bill Murray and "Sarge" in the latrine at basic (Sarge's office) the toilet bowl in the back left corner is installed backwards. (00:43:03 - 00:45:10)
Other mistake: During the graduation scene, if you watch as the platoons are marching by, you can see that several members are not in cadence as they should be. This also includes the men walking between the platoons that are supposed to be setting the cadence and they are out of step with the platoon in front and behind them.
Chosen answer: According to the movie's premise and plot, once they (the platoon) impressed the brass at graduation and was assigned to the Germany gig with the "EM-50", Sgt. Hulka would naturally be with them as he was their original commanding officer. Plus, it would be logical to suggest that he would personally assign himself the detail of overseeing his platoon as they were his to begin with.
CCARNI ★
He's assigned as a Drill Sergeant, that is his job. Once he was fully healed he would have been assigned another group of recruits.