  The pro-isomorphic zeta function of a finitely generated nilpotent group
$\Gamma$ is a Dirichlet generating function that enumerates finite-index
subgroups whose profinite completion is isomorphic to that of $\Gamma$. Such
zeta functions can be expressed as Euler products of $p$-adic integrals over
the $p$-adic points of an algebraic automorphism group associated to $\Gamma$.
In this way they are closely related to classical zeta functions of algebraic
groups over local fields.
  We describe the algebraic automorphism groups for a natural family of
class-$2$ nilpotent groups; these groups can be viewed as generalizations of
$D^*$-groups of odd Hirsch length. General $D^*$-groups, that is
`indecomposable' finitely generated, torsion-free class-$2$ nilpotent groups
with central Hirsch length $2$, were classified up to commensurability by
Grunewald and Segal. We calculate the local pro-isomorphic zeta functions for
our groups and obtain, in particular, explicit formulae for the local
pro-isomorphic zeta functions associated to $D^*$-groups of odd Hirsch length.
From these we deduce local functional equations; for the global zeta functions
we describe the abscissae of convergence and find meromorphic continuations. We
deduce that the spectrum of abscissae of convergence for pro-isomorphic zeta
functions of class-2 nilpotent groups contains infinitely many cluster points.
For instance, the global abscissae of convergence of the pro-isomorphic zeta
functions of $D^*$-groups of odd Hirsch length are determined and yield the
cluster point $6$.
