  The equivalence (or weak equivalence) classes of orientation-preserving free
actions of a finite group G on an orientable 3-dimensional handlebody of genus
g can be enumerated in terms of sets of generators of G. They correspond to the
equivalence classes of generating n-vectors of elements of G, where
n=1+(g-1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For
abelian and dihedral G, this allows a complete determination of the equivalence
and weak equivalence classes of actions for all genera. Additional information
is obtained for solvable groups and for the groups PSL(2,3^p) with p prime. For
all G, there is only one equivalence class of actions on the genus g handlebody
if g is at least 1+r(G)|G|, where r(G) is the maximal length of a chain of
subgroups of G. There is a stabilization process that sends an equivalence
class of actions to an equivalence class of actions on a higher genus, and some
results about its effects are obtained.
