 Deletion projection for a symmetric group

Delete $n$ from the cycle notation of a permutation in $S_n$ to obtain a permutation in $S_{n-1}$. Extend this set map linearly to ${\Pi }_n:\mathbb{C}{S}_n\to \mathbb{C}{S}_{n-1}$. It is an $S_{n-1}$-bimodule map and the identity on $\mathbb{C}{S}_{n-1}$, but is not in general an algebra homomorphism. In particular,For $n\ge 5$ it is the unique bimodule-equivariant set map $S_n\to S_{n-1}$; at $n=4$ another such map exists because $C_{S_3}(S_2)=S_2$.