Let $\phi$ be an automorphism of an abelian finite group, say $G$ and $s,t$ be the generating elements of $G$.
When we know the values $\phi(s), \phi(t)$, then we can find the value of any element under $\phi$. In other words, when we know where the generating elements are mapped, we know the entire automorphism.
Suppose $\theta$ is another automorphism of $G$. Suppose we don't know $\theta(s), \theta(t)$. But we know $\theta(\phi(s)), \theta(\phi(t)), \phi(s), \phi(t)$.
Knowing $\theta(\phi(s)), \theta(\phi(t))$, is like knowing the where the generating elements are mapped under the composition automorphism $\theta \phi$.
Then using this information how can I determine, $\theta$?
(Determine $\theta(s), \theta(t)$ ?)
Thanks a lot in advance.