  Mutant 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 their m-string satellites can have different Homfly polynomials
for m>2. We show that, under conditions of extra symmetry on the constituent
2-tangles, the directed m-string satellites of mutants share the same Homfly
polynomial for m<6 in general, and for all choices of m when 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 quantum sl(3)
invariants.
