There is a well-known morphism $S_4\to S_3$, obtained by having $S_4$ act on the three partitions of $4$ objects into $2+2$. Similarly, given any $n$, one can devise a morphism $S_n\to S_k$ for some $k$ by having $S_n$ act on the partitions of $n$ objects into $n_1+n_2+\ldots+n_\ell$. One can further endow some or all of the element of the partition with an order, for example with $\ell=1$ one has $S_n$ acting on the set of ordered sequences of size $n$, and gets the left action of $S_n$ on itself, which is a morphism $S_n\to S_{n!}$ (the largest one that is irreducible).

Is that, are something close to that, the complete list of all morphisms $S_n\to S_k$ (up to conjugacy, of course)? I assume the answer is well-known.

There is a well-known morphism $S_4\to S_3$, obtained by having $S_4$ act on the three partitions of $4$ objects into $2+2$. Similarly, given any $n$, one can devise a morphism $S_n\to S_k$ for some $k$ by having $S_n$ act on the partitions of $n$ objects into $n_1+n_2+\ldots+n_\ell$. One can further endow some or all of the element of the partition with an order, for example with $\ell=1$ one has $S_n$ acting on the set of ordered sequences of size $n$, and gets the left action of $S_n$ on itself, which is a morphism $S_n\to S_{n!}$ (the largest one that is irreducible).

Is that, are something close to that, the complete list of all morphisms $S_n\to S_k$ (up to conjugacy, of course)? I assume the answer is well-known.

Edit: the question was really naive, but I would like to know if some general information are nevertheless available on the Burnside ring of $S_n$ (which encodes the permutation representations of a finite group, but this is almost all I know).