  A real Lie algebra is said to be characteristically solvable if its
derivation algebra is solvable. We explicitly determine the moduli space of
left-invariant metrics, up to isometric automorphism, for $6$-dimensional
nilmanifolds whose associated Lie algebra is characteristically solvable. We
also compute the corresponding full isometry groups. For each left-invariant
metric on these nilmanifolds we compute the index and distribution of symmetry.
In particular, we find the first known examples of Lie groups which do not
admit a left-invariant metric with positive index of symmetry. As an
application we study the index of symmetry of nilsoliton metrics on
characteristically solvable Lie algebras. We prove that nilsoliton metrics
detect the existence of left-invariant metrics with positive index of symmetry.
