  The transpose top-$2$ with random shuffle (J. Theoret. Probab., 2020) is a
lazy random walk on the alternating group $A_n$ generated by $3$-cycles of the
form $(\star,n-1,n)$ and $(\star,n,n-1)$. We obtain the limit profile of this
random walk by comparing it with the random walk on $A_n$ generated by all
$3$-cycles. Our method employs a non-commutative Fourier analysis analogue of
the comparison method introduced by Nestoridi (Electron. J. Probab., 2024). We
also give the complete spectrum of the alternating group graph, thus answering
a question of Huang and Huang (J. Algebraic Combin., 2019).
