The operation which sends a finite set $S$ to the set of $k$-element subsets, $\binom{S}{k}$, gives rise to the $k$-th stable Hopf invariant. There is additional structure in this: the set $\binom{S}{k}$ has a canonical $k\!$ fold covering so the operation is better viewed as a map $$ QS^0 \to Q(B\Sigma_k)_+ $$ rather than as a map $$ Q S^0 \to QS^0 , $$ where for a based space $X$, the space $QX$ is $\Omega^\infty\Sigma^\infty X$ is the representing space for the stable homotopy of $X$, i.e., $\pi_j(QX) = \pi_j^{\text{st}}(X)$. So the operation induces a homomorphism $$ \pi_j^{\text{st}}(S^0) \to \pi_j^{\text{st}}((B\Sigma_k)_+) . $$
These operations satisfy certain axioms (Cartan Formula, compatibility with transfers, etc.). A good place to read about these operations is:
Segal, Graeme: Operations in stable homotopy theory. New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pp. 105–110. London Math Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974.