It is established in Whitten - Knot complements and groups together with the Gordon-Luecke theorem (that knot complements determine knot type) that the type of a prime knot is determined by the isomorphy type of its knot group.
In the book Charles Livingston - Knot Theory, the author uses surjective homomorphisms from knot groups into finite groups as knot invariants (i.e., two knot groups are nonisomorphic if one of them can be mapped onto a certain finite group and the other one can't). My question is:
If two prime knots are distinct, does that mean there is a finite group such that exactly one of them can be mapped surjectively into it?
Do all finite groups combined yield (as described above) a complete knot invariant for prime knots?