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 $[n+1,\ldots,n+k]$

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