A “book on group theory” with no reference to the Sylow theorems, Lagrange's theorem or Cayley's theorem?
Maybe this should be called “a book that illustrates gadgets that happen to have group structure”.
Maybe this should be called “a book that illustrates gadgets that happen to have group structure”.
This would be like asking the Nano developers how they plan to support Emacs-Lisp macros.
In other words, adjusting the title (to prevent wrong expectations) would be more productive than expanding the scope far beyond what the author(s) had in mind.
This book is about symmetry and group definitions, but it has no or very little details about generic properties of groups.
The previous comment is about a book that no one has written, that is a coloring book for students of a math major in the university, probably in the 2nd or 3rd year. In this (unwritten) book the idea is to study the general theorems about groups, and in each case give some visual examples to color and apply the theorem.
In particular, in this book they don't use $Z_2$ that is the usual notation in math (at least here). They use $C_2$ that is more usual in physics. And for the selection of the groups I guess the author has some interest in crystallography.