Are there hints of the exceptional nature of $S_6$ in its representation theory? $S_6$ is the unique symmetric group to have non-trivial outer automorphisms. It also has a triple cover.
Is it possible to see any hints of this exceptional behavior in its representations?
 A: Yes, at least the outer automorphism. Irreducible characters of the symmetric groups $S_n$ tend to come in pairs of the same degree, related by multiplying by the sign character. $S_6$ has $4$ irreducible characters of degree $5$, which split up into $2$ pairs under multiplying by the sign character. These pairs of characters are sent to each other by the exceptional outer automorphism of $S_6$; you can tell because the outer automorphism swaps the conjugacy classes $(12)$ and $(12)(34)(56)$ and this is how the two sets of characters are related as well.
I don't know how often it happens that $S_n$ has more than $2$ irreducible characters of the same degree. It doesn't happen for $S_5$, $S_4$, or $S_3$, but does happen for $S_7$, which apparently has $4$ irreducible characters of degree $14$.
I also don't know how the triple cover manifests if at all in the representation theory. I would wildly guess it affects the representation theory $\operatorname{mod} 3$ somehow based on some things I half-remember but I don't know any concrete results in this direction.
