Is there a classification of homomorphisms $S_n \to S_{n+k}$ for small $k$? Homomorphisms $B_n \to B_{2n}$ and $B_n \to S_{2n}$ have been classified in Chen–Kordek–Margalit - Homomorphisms between braid groups and Lin - Braids and permutations respectively. I am interested in the corresponding question for symmetric groups:  what are the homomorphisms $S_n \to S_{n+k}$ (up to conjugation)?
I know from Explicit description of all morphisms between symmetric groups. that such a classification is difficult in full generality—I am willing to restrict my attention to suitably small $k$. The answer to the linked question discusses the classification of maximal subgroups of $S_m$ isomorphic to $S_n$, but I do not see how this is directly useful, as the image of a map $S_n \to S_{n+k}$ needn't be maximal.
When $k = 1$ and $n + 1 \neq 6$, the only index $n+1$ subgroups of $S_{n+1}$ are point stabilizers. Thus, any non-cyclic map $S_n \to S_{n+1}$ is conjugate to the obvious inclusion.
When $k > 1$, we may combine* the identity map $S_n \to S_n$ with a sign representation $S_n \to \mathbb{Z}_2 \to S_{k}$ to produce a map $S_n \to S_{n+k}$ not conjugate to an inclusion. Following Chen–Kordek–Margalit, we might hope that these are all such homomorphisms (for small $k$).
Is anything known about these homomorphisms?
*In case it wasn't clear: this combination is found by letting $S_n$ act on $[1,\ldots, n]$ and $S_k$ act on $[n+1,\ldots,n+k]$
 A: As spin observed in a comment below, if you know all (conjugacy classes of) subgroups of $S_n$ of index up to $m$, then you can determine the equivalence classes of homomorphisms $\phi:S_n \to S_m$, because the subgroups tell you the possible actions on the orbits of the image of $\phi$. The subgroups of index up to $m=n^2$ are listed explicitly in Mikko Korhonen's answer to the the MSE post Large subgroups of Symmetric Group.
As an illustration, I will answer the question on the assumption that $k<n$.
For $n \ge 5$, a homomorphism from $S_n$ that is not injective has image of order $1$ or $2$, so we can restrict attention to injective maps $\phi:S_n \to S_{n+k}$.
From the MSE post referred to above we find that, for $n>6$, the only subgroups of index less than $2n$ are $S_n$, $A_n$, and the point stabilizers, which are isomorphic to $S_{n-1}$.
So if $S_n$ is acting faithfully on the set $\Omega := \{1,2,\dotsc,n+k\}$ with $k<n$, then there must be a single orbit $\Delta$ of length $n$ on which $S_n$ acts faithfully.
Since we are interested in classifying maps up to conjugation, we can assume that $\Delta = \{1,2,\dotsc,n\}$ and that the image of $\phi(g)$ restricted to $\Delta$ is $g$ for all $g \in S_n$.
Furthermore, the image of $\phi$ restricted to $\Omega \setminus \Delta$ has order $1$ or $2$.
So, $\phi(g)_{\Omega \setminus \Delta} = 1$ for $g \in A_n$, and for $g \in S_n \setminus A_n$, we have, up to conjugation,
$\phi(g)_{\Omega \setminus \Delta}$ can be
$1$, or $(n+1,n+2)$, or $(n+1,n+2)(n+3,n+4)$, etc., which gives a total of $\lfloor \frac{k+2}{2} \rfloor$ equivalence classes of injective homomorphisms $\phi$.
