If you have a $\phi$-invariant, normal subgroup $N$ (so $\phi(N)=N$) of a finite group $G$, for an $\phi$, then you get an induced automorphism of $G/N$ by $gN\mapsto \phi(gN)=\phi(g)N$. The order of the induced automorphism is a divisor of $\phi$.
If $\phi$ is a regular automorphism (only fixed point is the identity), then the induced automorphism is regular as well. Is the order of the induced automorphism equal to the order of $\phi$ if it is regular?