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 is a group, an automorphism of order of which is not an inner automorphism, pick a symbol and consider all elements , subject to if and only if , and . This way we obtain a larger group and group generated by cyclic group of order .

I have the following questions

1. What is the nature of the "a larger group " under discussion I mean the nature of the elements the operation.

2. As I understand this symbol that Herestein talks of abides to the binary operation of . Correct?

3. Is it correct to assume that where is the identity element.

4. I have difficulty imagining in but I guess I will come to that once I am clear on the behavior of as a group.

Thanks in advance.

~Kalyan.