Let's start with some family of algebraic structures of the same type indexed by the natural numbers, say the symmetric group $S_n$. Suppose that the axioms of this algebraic structure (in this case, groups) can be stated within the framework of first-order logic. In this way, we can consider a structure $M$ defined as follows: it contains $\mathbb{N}$, as well as all the $S_n$. Consider the complete theory $Th(M)$ in first-order logic (i.e. containing all the symbols of $M$ and a symbol for each relation on $M$ of any arity).

In this theory, there are relations $m, e, i$ that correspond to the relations of multiplication, identity, and inverse, and a relation $P$ such that $P(a,b)$ iff $a \in S_n$ and $b = n$. There are a bunch of axioms that are satisfied, e.g. if $P(a,b)$ then $P( i(a), b)$ and $m( a, i(a)) = e(b)$, etc., that correspond to the group laws. There is also a relation $isParameter$ that tells you whether the object in question is a parameter (i.e. a natural number).

So, in this way (am I misunderstanding this?) one can use an ultraproduct construction (or the compactness theorem applied to $Th(M)$ together with the collection of sentences $isParameter(c) \wedge c> 1 + \dots + 1$ where $c$ is some new constant symbol) to embed $M$ in a bigger structure $M'$ where there are parameters greater than all the standard elements of $\mathbb{N}$, i.e. where the parameters are (some version of) the hypernatural numbers. Since the group $S_n$ is nonempty for each standard $n$, it should follow by transfer that $S_n$ is also defined for infinite $n$. It must be a group by transfer.

Question:Is $S_{n}$ for infinite $n$ in any way related to the set of permutations of the interval from $1$ to $n$? If not, what can we say about it?

My hunch is that this probably isn't the case, because "every permutation is contained in $S_n$" sounds like a second-order statement and this is first-order logic we're dealing with. Nevertheless, I'm curious about what we can say about $S_{n}$ for infinite $n$, and whether we can deduce additional properties about $S_n$ for infinite $n$ from the known theory for finite $n$.