My professor made the following remark while teaching about group extensions:
We want to classify finite groups in a manner similar to the fact that every positive integer is uniquely a product of primes. The analogue of the equation $n = ab$ for integers is that you can sometimes "factor" a group $G$ into a normal subgroup $A$ and a quotient group $B = G/A$. You can also express this as an exact sequence of groups in the form:
$$1 \to A \to G \to B \to 1$$
But this is simply fancier notation for the exact same concept, that $A$ is a normal subgroup and $B = G/A$. Also, point of terminology, G is called an extension of $B$ by $A$.
(...)
At this moment, you should wonder a little about the possibility that $A$ has more than one complement in $G$. After all, a subspace $W$ of a vector space $V$ usually has many complements. All complements of $A$ are isomorphic to $B$, but their positions as subsets of $G$ can differ. So you could ask, how much can they differ as semidirect products? The answer is that sometimes the twisting homomorphism $\phi$ from $B$ to $\mathrm{Aut}(A)$ is exactly the same for different complements $B$, while sometimes $\phi$ changes moderately. You cannot radically change $\phi$ by changing the complement of $A$; it can only vary by automorphisms that come from conjugation in $A$. (These are called inner automorphisms.) In particular, if $A$ is abelian, then $\phi$ is unique.
Could someone please provide me a detailed explanation or proof of the fact highlighted in bold?
I'm not sure why $\phi$ can be modified only by inner automorphisms and not (say) by outer automorphisms too (on changing the complement of $A$). Moreover, why can't $\phi$ be some random homomorphism that isn't defined by a conjugation in $A$?
Why is $\phi$ necessarily unique when $A$ is abelian?
Here's the sequel to this question: Some clarifications required about the two extremes of general extensions (semi-direct products and central extensions)