  For a group $G$ first order definable in a structure $M$, we continue the
study of the "definable topological dynamics" of $G$. The special case when all
subsets of $G$ are definable in the given structure $M$ is simply the usual
topological dynamics of the discrete group $G$; in particular, in this case,
the words "externally definable" and "definable" can be removed in the results
described below.
  Here we consider the mutual interactions of three notions or objects: a
certain model-theoretic invariant $G^{*}/(G^{*})^{000}_{M}$ of $G$, which
appears to be "new" in the classical discrete case and of which we give a
direct description in the paper; the [externally definable] generalized Bohr
compactification of $G$; [externally definable] strong amenability. Among other
things, we essentially prove: (i) The "new" invariant $G^{*}/(G^{*})^{000}_{M}$
lies in between the externally definable generalized Bohr compactification and
the definable Bohr compactification, and these all coincide when $G$ is
definably strongly amenable and all types in $S_G(M)$ are definable, (ii) the
kernel of the surjective homomorphism from $G^*/(G^*)^{000}_M$ to the definable
Bohr compactification has naturally the structure of the quotient of a compact
(Hausdorff) group by a dense normal subgroup, and (iii) when $Th(M)$ is NIP,
then $G$ is [externally] definably amenable iff it is externally definably
strongly amenable.
  In the situation when all types in $S_G(M)$ are definable, one can just work
with the definable (instead of externally definable) objects in the above
results.
