Admission Impossible

I am confused about the usages of "admission"

1. "He nodded in (an) admission of guilt."

2. "There are always trouble spots in the college admission(s) process."

For 1., is "an" necessary/optional?

For 2., it seems that "admissions" means something different from "admission". Even when referring to the departments of schools that handles admitting new students, those departments are always called "admissions department", not "admission department".