Let $p$ be any prime. Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. Let $φ$ and $ψ$ be linear characters of $P_n$ and let $N$ be the normaliser of $P_n$ in $S_n$. In this article we show that the inductions of $φ$ and $ψ$ to $S_n$ are equal if, and only if, $φ$ and $ψ$ are $N$--conjugate. This is an analogue for symmetric groups of a result of Navarro for $p$-solvable groups.