  In part I it was shown that for each k>0 the generalized Sato-Levine
invariant detects a gap between k-quasi-isotopy of link and peripheral
structure preserving isomorphism of the finest quotient G_k of its fundamental
group, `functorially' invariant under k-quasi-isotopy. Here we show that
Cochran's derived invariant \beta^k, provided k>2, and a series of
\bar\mu-invariants, starting with \bar\mu(111112122) for k=3, also fall in this
gap. In fact, all \bar\mu-invariants where each index occurs at most k+1 times,
except perhaps for one occuring k+2 times, can be extracted from G_k, and if
they vanish, G_k is the same as that of the unlink.
  We also study the equivalence relation on links (called `fine
k-quasi-isotopy') generated by ambient isotopy and the operation of interior
connected sum with the second component of the (k+1)-th Milnor's link, where
the complement to its first component is embedded into the link complement. We
show that the finest quotient of the fundamental group, functorially invariant
under fine k-quasi-isotopy is obtained from the fundamental group by forcing
all meridians to be (k+2)-Engel elements. We prove that any group generated by
two 3-Engel elements has lower central series of length <6.

Geometric Topology;Group Theory