Multiplicity-free induced characters of symmetric groups

Let $n$ be a non-negative integer. Combining algebraic and combinatorial techniques, we investigate for which pairs $(G,\rho )$ of a subgroup $G$ of the symmetric group $S_n$ and an irreducible character $\rho$ of $G$ the induced character $\rho \text{\hspace{0.17em}⁣}{\uparrow }^{S_n}$ is multiplicity-free. As a result, for $n\ge 66$, we classify all subgroups $G\le S_n$ 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 $\rho$, assuming $n\ge 73$.