  Let $p\ge 5$ be a prime and let $n$ be a natural number. In this article we
describe the irreducible constituents of the induced characters
$\phi\big\uparrow^{\mathfrak{S}_n}$ for arbitrary linear characters $\phi$ of a
Sylow $p$-subgroup of the symmetric group $\mathfrak{S}_n$, generalising
earlier results of the authors. By doing so, we introduce Sylow branching
coefficients for symmetric groups.
