The monoid of all maps on $n$ letters is denoted $T_n$ and called the full transformation monoid. Your intuition is both right and wrong. The irreducible representations of $T_n$ are in bijection with irreducible representations of all symmetric groups of degree at most $n$. The character table is block upper triangular with diagonal blocks character tables of symmetric groups of degree at most $n$. Thus inverting the table, which is how you decompose characters into irreducibles, is a sort of generalized inclusion-exclusion. Putcha computed the character table, although I think Zuckermann stated the result without complete proof long ago. But this is only the semisimple part of the story.
The algebra of $T_n$ is not semisimple. For example, nobody knows what the projective indecomposables look like in general. The algebra is quasihereditary and so has finite global dimension. Putcha computed the quiver. Ponizovsky showed $T_3$ has finite representation type, Ringel showed $T_4$ has finite representation type and Putcha showed the representation type is infinite in all other cases. I think this part of the representation theory is no not completely controlled by symmetric groups.
