  The pro-isomorphic zeta function of a finitely generated nilpotent group is a
Dirichlet generating series that enumerates all finite-index subgroups whose
profinite completion is isomorphic to that of the ambient group. We study the
pro-isomorphic zeta functions of $\mathbb{Q}$-indecomposable $D^*$-groups of
even Hirsch length. These groups are building blocks of finitely generated
class-two nilpotent groups with rank-two centre, up to commensurability. Due to
a classification by Grunewald and Segal, they are parameterised by primary
polynomials whose companion matrices define commutator relations for an
explicit presentation. For Grunewald-Segal representatives of even Hirsch
length of type $f(t)=t^m$, we give a complete description of the algebraic
automorphism groups of associated Lie lattices. Utilising the automorphism
groups, we determine the local pro-isomorphic zeta functions of groups
associated to $t^2$ and $t^3$. In both cases, the local zeta functions are
uniform in the prime $p$ and satisfy functional equations. The functional
equations for these groups, not predicted by the currently available theory,
prompt us to formulate a conjecture which prescribes, in particular,
information about the symmetry factor appearing in local functional equations
for pro-isomorphic zeta functions of nilpotent groups. Our description of the
local zeta functions also yields information about the analytic properties of
the corresponding global pro-isomorphic zeta functions. Some of our results for
the $D^*$-groups associated to $t^2$ and $t^3$ generalise to two infinite
families of class-two nilpotent groups that result naturally from the initial
groups via `base extensions'.
