  The tree-level q-map assigns to a projective special real (PSR) manifold of
dimension $n-1\geq 0$, a quaternionic K\"{a}hler (QK) manifold of dimension
$4n+4$. It is known that the resulting QK manifold admits a
$(3n+5)$-dimensional universal group of isometries (i.e. independently of the
choice of PSR manifold). On the other hand, in the context of Calabi-Yau
compactifications of type IIB string theory, the classical hypermultiplet
moduli space metric is an instance of a tree-level q-map space, and it is known
from the physics literature that such a metric has an
$\mathrm{SL}(2,\mathbb{R})$ group of isometries related to the
$\mathrm{SL}(2,\mathbb{Z})$ S-duality symmetry of the full 10d theory. We
present a purely mathematical proof that any tree-level q-map space admits such
an $\mathrm{SL}(2,\mathbb{R})$ action by isometries, enlarging the previous
universal group of isometries to a $(3n+6)$-dimensional group $G$. As part of
this analysis, we describe how the $(3n+5)$-dimensional subgroup interacts with
the $\mathrm{SL}(2,\mathbb{R})$-action, and find a codimension one normal
subgroup of $G$ that is unimodular. By taking a quotient with respect to a
lattice in the unimodular group, we obtain a quaternionic K\"ahler manifold
fibering over a projective special real manifold with fibers of finite volume,
and compute the volume as a function of the base. We furthermore provide a
mathematical treatment of results from the physics literature concerning the
twistor space of the tree-level q-map space and the holomorphic lift of the
$(3n+6)$-dimensional group of universal isometries to the twistor space.
