You're correct that moving from the integers to the rationals does not fit, because the generalisation that inductive reasoning refers to is a generalisation of statements (or predicates) not of the objects themselves.
For example, you could generalise from the statement "all the even numbers above 3 we ever tried can be written as the sum of two primes" to "all of them can" - and that's an example of inductive reasoning. (There's there's no known deductive proof of this conjecture.)
Even numbers can be "generalised" to all numbers, but that's different to inductive reasoning. We don't move by inductive reasoning to "all whole numbers above 3 are the sum of two primes" because we find that 11 doesn't work.
Generalising generally vs inductive reasoning
"Generalising" the integers to the rationals is a superset relationship, which I can write very simply in maths notation, because it's like
The generalisation that inductive reasoning makes is:
I've not generalised A to B, I've generalised P from being true in all of A to being true throughout B.
In fact, you could say that the rationals are an example of a field, whereas the integers are only an example of a ring - a generalisation of a field. Mathematicians "generalised" (not inductive reasoning) from the number systems, matrices and other examples to make the abstractions of groups, rings and fields. Some theorems about fields can be generalised to rings, which would be an example of inductive reasoning were it not for the fact that mathematicians don't admit inductive reasoning as proof, and proved these generalisations deductively!
(Proof by induction is, perhaps confusingly, an example of deductive reasoning, but is often something you do after having used inductive reasoning to conclude that you might want to prove your statement.)
A deep understanding of inductive reasoning isn't necessary for good abstractions, but good abstractions result from deep understanding of diverse examples.
Abstraction is a form of generalisation, and supersets are a form of generalisation, but inductive reasoning specifically means generalising statements from examples to every example.
All inductive reasoning is generalisation, but not all generalisation is inductive reasoning.