Noether's problem for $A_6$ and $A_7$

We prove that $B_k{A}_6$ is stably rational over every field $k$ in which $2$ and $-3$ are nonzero squares. Combining this with a theorem of Plans, we deduce that $B_k{A}_7$ is stably rational over every field $k$ of characteristic zero in which $2$ and $-3$ are squares.