Automorphisms and Extensions of Cyclic Groups

Question

I can't figure out what question III.4.2 from Hilton & Stammbach's book A Course in Homological Algebra is asking:

Classify the extension classes $[E]$ given by: $$ \mathbb Z_m \rightarrowtail E \twoheadrightarrow \mathbb Z_n$$ under automorphisms of $\mathbb Z_m$ and $\mathbb Z_n$.

My understanding is that, fixing an extension $E$, work out which other extensions appear as those induced by automorphisms of either $\mathbb Z_m$ or $\mathbb Z_n$. However,

1. I am unsure if my understanding is correct,
2. I can't work out an a way to answer the exercise the way I understood it.