Mystified by construction of "group extension" $\{G, T\}$ in Herstein's "Topics in Algebra" On page 69 of my (ancient) copy of his Topics in Algebra, I. N. Herstein describes a baffling construction by way of example, and moves on without further comment (FWIW, note that Herstein applies automorphisms "on the right"; i.e. his $xT$ is what others would write as $Tx$ or even $T(x)$):

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 $x^ig$, $i = 0, \pm 1, \pm 2,\dots,\; g\in G$ subject to $x^i g = x^{i^{\,\prime}}\!\!g^{\,\prime}$ if and only if $i \equiv i^{\,\prime} \!\!\! \mod r, g = g^{\,\prime}$ and $x^{-1}g^i x = g\,T^i$ for all $i$.  This way we obtain a larger group $\{G, T\}$; $G$ is normal in $\{G, T\}$ and $\{G, T\}/G\approx$ group generated by $T=$ cyclic group of order $r$.

OK, I get it (although just barely) that this $\{G, T\}$ is a group, etc., but what's the point?
I remember being similarly mystified, a long time ago, when I first came across the construction of a quotient group.  Now, with a lot more exposure to this sort of thing under my belt, it is not such a mystifying concept...  So, given this experience, I now have to wonder if the construction above is pointing at some workhorse maneuver in algebra.  Is it just some idiosyncratic hiccup of the author's, one that can safely be ignored?  Or is it worth my while to try to understand it better?  And if so, how?  When learning about quotient groups, it was very helpful for me to see how quotient groups generalized the notion of a projection of vector space onto a subspace.  Is there a similarly illuminating illustration of what the construction above is aiming at?
 A: The group $\{G,T\}$ constructed above is called the semidirect product of $G$ by $\langle T \rangle$, denoted $G \rtimes C_r$.  In the example given in the text, starting with an automorphism of $C_7$ of order 3, a nonabelian group of order 21 is constructed, and this group is the semidirect product $C_7 \rtimes C_3$.  
Semidirect products are a generalization of direct products, but unlike with the direct product, in a semidirect product only one of the two factors is required to be a normal subgroup.  Semidirect products are a nice way to construct nonabelian groups. An example of a group that can be described as a semidirect product but not as a direct product (of two smaller groups) is the dihedral group $D_6$ (i.e. $S_3$), which can be described as $C_3 \rtimes C_2$.  (If $D_6$ were also a direct product, the two factors would have to be abelian, which means $D_6$ would also be abelian, a contradiction.)  
You can learn about semidirect products in the algebra text by Dummit and Foote.
A: For getting a feeling for group extensions, understanding the semidirect product is certainly one part. If you then also understand the wreath product, you are prepared for the Kaloujnine-Krasner Theorem, which at least covers one part of the "story".
But the "story" would be misleading if you haven't met the concept of a central extension, like illustrated by the quaternion group. This shows a connection to a cohomology group. I haven't yet managed to "fully" understand group cohomology, but I think it is important to realize that "fully" understanding the group extension problem will also require a fair amount of understanding of group cohomology.
