I.N. Herstein says,
Generally, if $G$ is a group, $T$ an automorphism of order $r$ of $G$ which is not an inner automorphism, pick a symbol $x$ and consider all elements $x^ig$, $i=0,+1,+2,...-1,-2..,$ $g \in G$ subject to $x^ig=x^{i'}g'$ if and only if $i=i'mod r, g=g'$ and $x^{-1}g^ix=gT^i$ for all $i$. This way we obtain a larger group $\{G,T\}$...
This is the first time I am studying group theory in particular and abstract algebra so formally in general.
Naturally, even though I have followed the book closely up until this point so that I am able to understand what the author says, I do not feel satisfied at all.
Is there a more inisghtful/less adhoc/more natural way of understanding how automorphisms of a group enable creation of larger groups?
Edit: Note I am particularly asking for automorphisms which are not inner and for which the above quoted construction is used. The case of inner automorphism (and how the group created is huge!) is something (I think) I understand.