I'm dealing with the following problem in Isaacs Finite Group Theory [4D.3], I would appreciate if you could help:
Problem: Let $A$ act via automorphisms on $G$ , where $(\vert G \vert, \vert A \vert ) = 1.$ Suppose that $A$ acts trivially on every $A$-invariant proper subgroup of $G$, but that the action of $A$ on $G$ is nontrivial. Show that $G$ is $p$-group for some prime $p.$
My attempt: For a contradiction, assume that $G$ is not a $p$-group. Using Glauberman Lemma, we can see that there exists an $A$-invariant Sylow $p$-subgroup of $G$ for each prime. Due to hypothesis, for any $A$-invariant Sylow $p$-subgroup $P$, it must be $[P,A]=1$. That is each such a Sylow $p$-subgroup is contained in $C_G(A)$. From the hypothesis again, we know that $C_G(A) \neq G.$ How can I get the desired result from this point?