If you need to check your work, here are trusted sources:
When asked to find the kernel of an action, remember it is the intersection of all stabilizers: Section 4.3: Conjugacy Classes and the Class Equation This is where the algebra gets "computational." The Center ( dummit foote solutions chapter 4
– Explores the Class Equation , conjugacy classes, and centralizers. 4.4: Automorphisms – Discusses the group of automorphisms and inner automorphisms . If you need to check your work, here
(e.g., Section 4.3, Exercise 5), I can walk you through the proof step-by-step or explain the underlying logic! dummit foote solutions chapter 4