It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia:
The purpose of defining a group homomorphism as it is, is to create functions that preserve the algebraic structure.
Now, my default notion of structure is the one that holds that isomorphisms are structural identities---two isomorphic mathematical objects have the same structure. But that notion of structure requires preserving not just the operations on the elements of the domain but also requires the two sets of objects be equipollent.
But homomorphisms aren't, in general, reversible---or else they'd be isomorphisms. Since homomorphisms don't preserve size, in what sense does it preserve "structure"? Is there any more to the notion of structure here than the simple definition of algebraic structure as a set endowed with certain operations?
Toy example illustrating the problem: suppose you have a group homomorphism between groups $G$ and $H$ (but no isomorphism). Suppose also that you have an isomorphism between $G$ and some further group $J$. Since there is a homomorphism between $G$ and $H$, the structure in $G$ is preserved in $H$. Since there is an isomorphism between $G$ and $J$ they have the same structure. But presumably if $H$ preserves the structure of $G$ then they have the same structure. That can't be true, though, since they aren't isomorphic. It seems then that either "structure" means two different things when discussing isomorphisms and homomorphisms, or "preserves" doesn't mean that the homomorphic sets have the same structure (and then I have no idea what "preserves" actually means in this context).
I'm clearly confused about something. I'm just not sure what.
TL;DR What is the difference between the notions of "structure" when you say that isomorphic structures have the same structure and when you say that a group homomorphism preserves the group structure?