  Any knot in a solid torus, called a pattern or satellite operator, acts on
knots in the 3-sphere via the satellite construction. We introduce a
generalization of satellite operators which form a group (unlike traditional
satellite operators), modulo a generalization of concordance. This group has an
action on the set of knots in homology spheres, using which we recover the
recent result of Cochran and the authors that satellite operators with strong
winding number $\pm 1$ give injective functions on topological concordance
classes of knots, as well as smooth concordance classes of knots modulo the
smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite
operators yields a characterization of surjective satellite operators, as well
as a sufficient condition for a satellite operator to have an inverse. As a
consequence, we are able to construct infinitely many non-trivial satellite
operators 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 a
potentially exotic $S^3\times [0,1]$) for all knots K; we show that these
satellite operators are distinct from all connected-sum operators, even up to
concordance, and that they induce bijective functions on topological
concordance classes of knots, as well as smooth concordance classes of knots
modulo the smooth 4--dimensional Poincare Conjecture.
