Let $G$ be a nontrivial finite group. In his book "Finite Group Theory", M. Isaacs proves the following two results:
Corollary 3.3: Let $\sigma \in \operatorname{Aut}(G)$. Then, $o(\sigma) < |G|$.
Corollary 3.4: Let $P$ be an abelian $p$-subgroup of $\operatorname{Aut}(G)$ and suppose $p \nmid |G|$ . Then, $|P| < |G|$.
This lead me to the following question:
Question: is the same result (3.4) true when $p \mid |G|$?
I'm guessing "no", but I know very few concrete examples of automorphism groups of finite groups. If $G$ is cyclic, for instance, $|\operatorname{Aut}(G)| = \phi(n) < n$, meaning the question is automatically true.
If $G$ is dihedral, I found $\operatorname{Aut}(G) \simeq C_n \rtimes C_n^\times$, where $|G| = 2n$ and the action is $x^p \cdot x^q = x^{pq}, x^p \in C_n^\times, x^q \in C_n$, which has order $n \phi(n)$; in particular, for cases like $n = 8$, it seems like I could find a counter-example, but I'm still struggling a bit to find abelian subgroups of this semidirect product, since I'm a bit new to these groups...
EDIT: As has been suggested in the comments, the second question has been moved to another post.
Any help is appreciated! Thanks in advance!