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Any knot in a solid torus, called a pattern or satellite operator, acts onknots in the 3-sphere via the satellite construction. We introduce ageneralization of satellite operators which form a group (unlike traditionalsatellite operators), modulo a generalization of concordance. This group has anaction on the set of knots in homology spheres, using which we recover therecent result of Cochran and the authors that satellite operators with strongwinding number $\pm 1$ give injective functions on topological concordanceclasses of knots, as well as smooth concordance classes of knots modulo thesmooth 4--dimensional Poincare Conjecture. The notion of generalized satelliteoperators yields a characterization of surjective satellite operators, as wellas a sufficient condition for a satellite operator to have an inverse. As aconsequence, we are able to construct infinitely many non-trivial satelliteoperators P such that there is a satellite operator $\overline{P}$ for which$\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in apotentially exotic $S^3\times [0,1]$) for all knots K; we show that thesesatellite operators are distinct from all connected-sum operators, even up toconcordance, and that they induce bijective functions on topologicalconcordance classes of knots, as well as smooth concordance classes of knotsmodulo the smooth 4--dimensional Poincare Conjecture.

Satellite operators as group actions on knot concordance