I think the 1921 book Primitive Groups: Part I by William Manning is an interesting case. It has a heavy focus on permutation groups. In fact, all the usual definitions like subgroup, isomorphism, cyclic, order, Abelian, etc. are introduced in the context of permutation groups. Then it goes on to introduce groups of linear substitutions called "linear groups," and it states that most of the theorems related to permutation groups also hold for linear groups, and the term "group" can mean either one depending on the context. This appears similar to how, for example, in Axler's Linear Algebra Done Right, all fields are taken to be $\mathbb{R}$ or $\mathbb{C}$. Unfortunately, I can't find Part II of Manning's book, so I'm not sure if he ever even mentions that there are more general groups, unlike Axler who does acknowledge that there are other fields, with the exception of this line from the preface:

Moreover, since any "abstract" group of finite order is isomorphic to some group of permutations, it would seem that sufficient generality can be attained if the phraseology of the abstract theory is ignored, as is done in this book.

Bonus. This was the first book to use the word "homomorphism" for groups.