This article gives sharp estimates for the mixing time of the Burnside process for Sylow $p$-double cosets in the symmetric group $S_n$. This process is a Markov chain on $S_n$ which can be used to uniformly sample Sylow $p$-double cosets. The analysis applies when $n = pk$ with $p$ prime and $k < p$. The main result describes the limit profile of the distance to the stationary distribution as $p$ goes to infinity. From the limit profile, we get the following two corollaries. First, if $k$ remains fixed as $p \to \infty$, then order $p$ steps are necessary and sufficient for mixing and cut-off does not occur. Second, if $k \to \infty$ as $p \to \infty$, then cut-off occurs at $p \log k$ with a window of size $p$. The limit profile is derived from explicit upper and lower bounds on the distance between the Burnside process and its stationary distribution. These non-asymptotic bounds give very accurate approximations even for $p$ as small as 11.