  We show that finite index subgroups of the handlebody group are rigid in
their ambient mapping class group: any injective map of a finite index subgroup
of the genus $g$ handlebody group into the genus $g$ mapping class group is
conjugation by a mapping class group element.
  On the other hand, we construct an injection of the genus $g$ handlebody
group into a genus $h>g$ mapping class group which is not conjugate into a
handlebody group.
