Let $n$ be a non-negative integer. Combining algebraic and combinatorial techniques, we investigate for which pairs $(G,ρ)$ of a subgroup $G$ of the symmetric group $S_n$ and an irreducible character $ρ$ of $G$ the induced character $ρ\!\uparrow^{S_n}$ is multiplicity-free. As a result, for $n\geq 66$, we classify all subgroups $G\leq S_n$ which give rise to such a pair. Moreover, for the majority of these groups $G$ we identify all the possible choices of the irreducible character $ρ$, assuming $n\geq 73$.