(Soft Question) What kinds of properties are transferred by isomorphisms?

Mathematical logic (specifically model theory) provides a partial answer. Let $M$ and $N$ be structures for a first-order language $L$. $M$ and $N$ are elementarily equivalent if every closed formula satisfied by one is satisfied by the other. $M$ and $N$ are isomorphic if there is a 1-1 map between $M$ and $N$ that preserves all the relations and functions mentioned in the signature of $L$. Theorem: if $M$ and $N$ are isomorphic, then they're elementarily equivalent. See, say Marker Model Theory: An introduction, §1.1, or Hodges A Shorter Model Theory, §1.2.

I think this serves as a reasonable candidate for "a general theorem that all such properties/objects are preserved by isomorphisms in the category we're working in".

I say a partial answer, because choosing the language in each case remains an issue. Let me elaborate for your example of groups. We want to show that being a subgroup, or a normal subgroup, or the center, is preserved by isomorphisms, all in one shot. For $L$, we include the following in its signature: the constant symbol 1, the function symbols $\cdot,{}^{-1}$, and a unary relation symbol $S$ for the subset under discussion. (There are other signatures that would also serve.) Here are the closed formulas that express "$S$ is a subgroup", etc. I'm going to be a bit sloppy for increased readability, using juxtaposition for the operation and omitting parentheses. Also, when I write "$S$ is a subgroup" in the second two bullets, just imagine the first bullet being repeated in full.

• $S(1)\wedge\forall x\forall y[S(x)\wedge S(y)\rightarrow S(x^{-1})\wedge S(xy)]$
• $S$ is a subgroup and $\forall x\forall y[S(x)\rightarrow S(y^{-1}xy)]$
• $S$ is a subgroup and $\forall x[\forall y(yx=xy)\rightarrow S(x)]\wedge \forall x[S(x)\rightarrow\forall y(yx=xy)]$

So if $M$ and $N$ are isomorphic, then $M$ satisfies one of these formulas if and only if $N$ does—that's what elementary equivalence says. And if $M$ and $N$ are isomorphic groups, then the subsets defined by the relation symbol $S$ correspond, and therefore one is a subgroup (or normal, or the center, or anything expressible by a closed formula in this language) if and only if the other is.

If you're familiar with first-order logic, you'll be aware of various hurdles to overcome. For example, to define "commutator subgroup" with a closed formula, you'd need to expand the language to allow for sequences of arbitrary finite length, since the commutator subgroup is generated by the commutators. That means incorporating $\mathbb{N}$ into the structure in some manner. I don't mean that $\mathbb{N}$ would be a subset of the group, rather that the structure would be (implicitly) an ordered tuple $(G,\mathbb{N},\ldots)$. For "derived series" you'd need to expand the language some more. But all these obstacles can be mastered with standard techniques.

A fuller answer would discuss connecting the category theory with the model theory. I plead limitations of both space and my expertise.

I would argue that a "group theoretic property" or a "topological property" etc. is precisely defined to be a property that is invariant under group isomorphism, topological isomorphism (also called "homeomorphism" : as pointed out in the comments, there is only one notion of isomorphisms, it just so happens that for algebraic structures, there are equivalent formulations using bijections, i.e. isomorphisms in $\mathbf{Set}$), etc.

In this sense, the answer is tautological : group theoretic properties are preserved under isomorphism...because they are.

Of course that's not a satisfying answer, because this doesn't reduce the amount of proofs we have to do (we still have to prove that such property is preserved under isomorphism to prove that it's a group theoretic property).

But the advantage of taking this point of view is that it comes with a natural way of checking that something is, in fact, a group theoretic property (I'm using the example of groups here because it's easier to just use one example), so it shifts the focus on something else, and that something else is easier to make sense of.

Indeed, to check that something is preserved under isomorphism, that is, is a group theoretic property, it suffices to check that it can be defined internally to the category of groups.

For instance, "an element of order $\mid n$ of $G$" can be defined as a morphism $\mathbb Z/n \to G$; and an element of order $n$ as such a morphism that cannot be factored as $\mathbb Z/n\to \mathbb Z/d \to G$ for any $d<n$ (or you could say "an element of order $\mid n$ which is a monomorphism". Or you could say that an element is a morphism $\mathbb Z\to G$ and that it has order $n$ if and only if it can be factored as $\mathbb Z\to\mathbb Z/n\to G$ and no lower $d$, for instance.

Or else, an abelian group can be defined to be an object that admits a "group object" structure in the category of groups (this point of view is actually helpful in other regards), so it's invariant under isomorphism as well.

There are various ways of seeing that such and such definition can be defined categorically, but in the end it always allows you to see that it is invariant under isomorphism.

The reason is that properties that are defined internal to a category are invariant under isomorphism. To get a feel of why this is true, you may want to check out my other answer here, which attempts to explain that.

Let me add that, regardless of the philosophical question of whether something is a group theoretic property, or what that even means, the idea of expressing various concrete notions categorically can be extremely interesting.

Just to give an example : over a ring $R$, there's a notion of "finitely presented module". Now this is defined purely in terms of arrows and so on, so it's easy to see that it's invariant under isomorphism. But in fact, more is true : you can define it internal to the category of $R$-modules without using any specific $R$-module : finitely presented $R$-modules are precisely the compact objects of that category. Now the notion of a compact object is purely categorical (so it doesn't even refer to $R$-modules), and so it's transported along equivalences of categories. This can be helpful in setting up the bases of Morita theory.

This is one categorical level higher ("invariant under equivalence of categories"), so it's not entirely relevant to your question, but it shows that the more you can define things categorically, the more invariant they become; so it's a good argument in favour of the point of view I've tried to portray here.

But, as I pointed out (if I recall correctly) in my other answer, coming up with a precise (and useful !!) statement about this sort of thing, that applies in all contexts (the accepted answer's statement is certainly simple and precise; unfortunately it only applies in cases where you're dealing with categories of first order structures on a certain language - of course you can expand to higher orders etc. but it will nonetheless be limited) would actually be very difficult; and in the end, knowing what kind of things are invariant under isomorphism and what kind of things aren't is mostly a matter of experience.

You just know that being of order $n$ is preserved under an isomorphism; whereas $\pi\in G$ isn't. These things become obvious with experience - and sometimes, it is a problem because sometimes our intuition fails. For instance, sometimes you forget that things are invariant under isomorphism in another category, but might not be in the category you're actually considering.

I've never seen a blatantly wrong example though, of something that you would be convinced is preserved under isomorphism, even if you thought about it for a long long time; but that actually isn't. I think it's one of the most robust non-precise notions there is.