You can refine this. Let's take $k=2$ to give the idea. To a based set $X$ you can associate $(X\wedge X)/X$, a based set with free action of $\Sigma_2$. This leads to an operation going from stable homotopy of $S^0$ to stable homotopy of $B\Sigma_2$, such that when followed by transfer it gives the difference between the identity and squaring.

This leads to a proof of the Kahn-Priddy Theorem, a proof due to Kahn and Priddy I believe.

Waldhausen adapted the same idea to prove an important result about his $A(X)$, the "vanishing of the mystery homology theory". (That's why I know about it.) This involved extending the construction of these operations from the algebraic $K$-theory of sets to the algebraic $K$-theory of spaces.

You can refine this. Let's take $k=2$ to give the idea. To a based set $X$ you can associate $(X\wedge X)/X$, a based set with free action of $\Sigma_2$. This leads to an operation going from stable homotopy of $S^0$ to stable homotopy of $B\Sigma_2$, such that when followed by transfer it gives the difference between the identity and squaring.

This leads to a proof of the Kahn-Priddy Theorem, a proof due to Kahn and Priddy I believe. (EDIT: No, I guess it was Segal.)

Waldhausen adapted the same idea to prove an important result about his $A(X)$, the "vanishing of the mystery homology theory". (That's why I know about it.) This involved extending the construction of these operations from the algebraic $K$-theory of sets to the algebraic $K$-theory of spaces.