Kaluzhnin's theorem says that if $G$ is a group and $H \leq \operatorname{Aut}G$ acts trivially on each step of the normal series $1 = G_0 \leq G_1 \leq \ldots \leq G_n = G$, then $H$ has nilpotency class at most $n-1$.
I see how to prove this, and I am sure I've encountered this theorem in textbooks. But here's the catch: these textbooks were definitely in Russian. What is the standard English reference for this theorem?
I usually find basic results like this in Robinson or Rotman, but not this time. Even more astonishingly, I had no luck with Google either! Does this theorem have some other name that is more popular?
I think that you can find your answer in page 113 of this book(Theorem 16.3.1) "M. I. Kargapolov and Yu.I. Merzlyakov. General group theory, Moscow, 1972. English translation: Fundamentals of the theory of groups, translated from the second Russian edition by Robert G. Burns. Graduate Texts in Mathematics, 62. Springer-Verlag, New York, Berlin, 1979."