Translation request: theorem concerning automorphisms of $p$-groups I want to translate the following theorem from German to English:

5.12 Satz. Sei $\mathfrak{P}$ eine $p$-Gruppe und $\alpha$ ein Automorphismums von $\mathfrak{P}$ von su $p$ teilerfremdder Ordnung.*
  
  
*
  
*Für $p\gt 2$ lasse $\alpha$ alle Elemente der Ordnung $p$ von $\mathfrak{P}$ fest.
  
*Für $p\gt 2$ lasse $\alpha$ alle Elemente der Ordnungen $2$ und $4$ von $\mathfrak{P}$ fest.
Dann ist $\alpha =1$.

I used google translator and get the following:
Let B be a p-group and S be an automorphism of B of order coprime to p.
a) For p> 2, let S fix all the elements of B of order p.
b) for p = 2, let S fix all elements of orders 2 and 4 of B.
Then S = 1
I found the following translation on some research

For a finite $p$ group $P$, we write
  $$\Omega(P) = \left\{\begin{array}{ll}
\Omega_1(P) &\text{if }p\gt 2\\
\Omega_2(P) &\text{if }p=2
\end{array}\right.$$
  where
  $$\Omega_i(P) = \Bigl\langle x\in P \;\Bigm|\; |x| \bigm| p^i\Bigr\rangle.$$
Lemma 2.1 ([6] Kap. IV, Satz 5.12]). Suppose that a finite $p'$-group $A$ acts by automorphisms on a finite $p$-group $P$. If $A$ acts trivially on $\Omega_1(P)$ for $p\neq 2$ or on $\Omega_2(P)$ for $p=2$,then $A$ acts trivially on $P$.

I want to know if the last translation is true because I am not sure that if an automorphism comes from a $p^{'}$-group, then this automorphism is of order coprime to $p$.
 A: A better translation of the Theorem is:

Theorem 5.12. Let $P$ be a $p$-group, and let $\alpha$ be an automorphism of $P$ of order coprime to $p$.



*

*If $p\gt 2$ and $\alpha$ fixes all elements of order $p$; or

*If $p=2$ and $\alpha$ fixes all elements of orders $2$ or $4$;
then $\alpha$ is the identity.
The quoted lemma is equivalent to this result.
To see that the lemma follows from "Theorem 5.12", note that the fact that $A$ acts by automorphisms on $P$ is equivalent to saying that we have a group homomorphism $A\to \mathrm{Aut}(P)$. The conditions given on $A$ imply that if $\alpha\in A$, then the image of $\alpha$ in $\mathrm{Aut}(P)$ has order coprime to $p$ and satisfies the hypothesis of Lemma 5.12, hence $\alpha$ is the identity. Thus, the image of $A$ is trivial, so $A$ acts trivially on $P$.
Conversely, if the quoted Lemma is true, then Theorem 5.12 follows. Let $\alpha$ be an automorphism of $P$ of order coprime to $p$ that fixes each of the elements of order $p$ if $p\gt 2$, and the elements of orders $2$ and $4$ if $p=2$. Let $A=\langle \alpha\rangle$, viewed as a subgroup of $\mathrm{Aut}(P)$. Then $A$ acts on $P$ and satisfies the hypothesis of the lemma, hence acts trivially. In particular, $\alpha$ acts trivially; i.e., $\alpha$ is the identity automorphism. This proves "Theorem 5.12". 
