  We consider the hyperelliptic handlebody group on a closed surface of genus
$g$. This is the subgroup of the mapping class group on a closed surface of
genus $g$ consisting of isotopy classes of homeomorphisms on the surface that
commute with some fixed hyperelliptic involution and that extend to
homeomorphisms on the handlebody. We prove that the logarithm of the minimal
dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the
hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means
that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the
subgroup of genus $g$ in question is the same as that of the ambient mapping
class group of genus $g$. We also determine finite presentations of the
hyperelliptic handlebody groups.
