Obtaining a larger group from a given group

[Excerpt from Topics in Abstract Algebra Second Edition by I.N.Herstein pg 69]

This is what Herstein talks of verbatim:

Generally, if $\displaystyle G$ is a group, $\displaystyle T$ an automorphism of order $\displaystyle r$ of $\displaystyle G$ which is not an inner automorphism, pick a symbol $\displaystyle x$ and consider all elements $\displaystyle x^ig$, $\displaystyle i= 0, \pm1, \pm2,.... g \in G $ subject to $\displaystyle x^ig = x^i^'$$\displaystyle g^'$ if and only if $\displaystyle i \equiv i^'$$\displaystyle mod r$, $\displaystyle g = g^'$ and $\displaystyle x^{-1}g^{i}x = gT^{i} \forall i$. This way we obtain a larger group $\displaystyle \{G,T\}$ and $\displaystyle \{G,T\}/G \approx$ group generated by $\displaystyle T = $cyclic group of order $\displaystyle r$.

I have the following questions

1. What is the nature of the "a larger group $\displaystyle \{G,T\}$" under discussion I mean the nature of the elements the operation.

2. As I understand this symbol $\displaystyle x$ that Herestein talks of abides to the binary operation of $\displaystyle G$. Correct?

3. Is it correct to assume that $\displaystyle x^r = e$ where $\displaystyle e$ is the identity element.

4. I have difficulty imagining $\displaystyle G$ in $\displaystyle \{G,T\}$ but I guess I will come to that once I am clear on the behavior of $\displaystyle \{G,T\}$ as a group.

Thanks in advance.

~Kalyan.