AbstractMutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general theirm-string satellites can have different Homfly polynomials form> 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directedm-string satellites of mutants share the same Homfly polynomial form< 6 in general, and for all choices ofmwhen the satellite is based on a cable knot pattern.We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantumsl(3) invariants.