rt.representation theory - Explicit description of all morphisms between symmetric groups.

Bret Benesh and Ben Newton determined all pairs $(m,n)$ such that $S_m$ contains a maximal subgroup isomorphic to $S_n$. They are either $(n+1,n)$ with the obvious inclusion (or mapping $S_5$ into the image of a point stabilizer under the outer automorphism of $S_6$); $(binom{n}{k},n)$, coming from the action of $S_n$ on the subsets of $k$ elements of ${1,2,ldots,n}$; and $((kr)!/(r!)^k k!, kr)$ with $1lt k,r$, with $S_{kr}$ acting on the the right cosets of a maximal subgroups of the wreath product $S_kwr S_r$. This appears in A classification of certain maximal subgroups of symmetric groups, J. Algebra 304 (no. 2) pp. 1108-1113, MR2265507.

Bret later also determined all pairs $(m,n)$ such that $S_m$ has a maximal subgroup isomorphic to $A_n$; such that $A_m$ has a maximal subgroup isomorphic to $S_n$; and such that $A_m$ has a maximal subgroup isomorphic to $A_n$. This appears in the book Computational Group Theory and the Theory of Groups, Contemporary Mathematics 470 (L-C Kappe, R. F. Morse, and me as editors), AMS 2008; the paper is A classification of certain maximal subgroups of alternating groups, pp. 21-26, MR2478411.

As pointed out by Jack, this does exhaust all possible embeddings of $S_n$ into $S_k$ (presumably you are okay with the maps that are not embeddings...)