A cone with a swept fin in high-speed flow produces a complex, three-dimensional boundary layer with a system of steady vortices in addition to crossflow. Experiments in the Boeing/AFOSR Mach 6 Quiet Tunnel on a 7° half angle cone with a 70° swept fin indicate the onset of transition on the cone in the region coinciding with the horseshoe vortex for some freestream conditions. The present work aims to identify the mechanisms of perturbation growth in the fin-cone boundary layer. We use the Navier-Stokes equations to compute a high-fidelity steady base flow at a unit Reynolds number of 7.3 × 106 m−1. We introduce azimuthally invariant perturbations at the wall upstream of the fin and investigate the response of the boundary layer as the perturbation wavepacket travels downstream. The boundary layer is thin on the downwash side of the horseshoe vortex and thicker on the upwash side. Low frequency (150-250 kHz) acoustic instabilities therefore amplify away from the fin whereas high frequency (250-350 kHz) acoustic instabilities amplify closer to the fin. There exists a region on the cone in the downwash of the fin leading edge vortex where the boundary layer is thin and can support the growth of high frequency instabilities far downstream relative to an axisymmetric cone. The horseshoe vortex also supports an instability localized away from the wall, which is different from the second mode instability. The results provide insight into the linear mechanisms that cause growth of small amplitude perturbations and are expected to help identify transition mechanisms in the presence of high amplitude perturbations.