Consider that a group $A$ acts by automorphism on a finite group $G$. If this action is coprime, i.e. $\gcd(|A|,|G|)=1,$ can we affirm that this action is fixed point free, i.e. $C_G(A)=1$?
I tried to think about the order of the automorphism and the order of na element of $G$, but it doesn’t work.
As Derek Holt pointed out in the comments, the action may have fixed points, even if we don't count the identity. For example, take $G$ to be the quaternion group of order 8, and let $A = \mathbb Z/3\mathbb Z$ act by cycling $i, j$, and $k$. This fixes the central element $-1$.