Homomorphisms $B\_n \to B\_{2n}$ and $B\_n \to S\_{2n}$ have been classified in [Chen–Kordek–Margalit - Homomorphisms between braid groups](https://people.math.gatech.edu/%7Edmargalit7/papers/bn2bn.pdf) and [Lin - Braids and permutations](https://arxiv.org/abs/math/0404528) 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.](https://mathoverflow.net/questions/36452/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 $[n+1,\ldots,n+k]$



Is there a classification of homomorphisms $S_n \to S_{n+k}$ for small $k$?
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](https://math.stackexchange.com/questions/3536888).


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 $\phi$.


