  Given a conical affine special K\"{a}hler (CASK) manifold together with a
compatible mutually local variation of BPS structures, one can construct a
quaternionic-K\"{a}hler (QK) manifold. We call the resulting QK manifold an
instanton corrected c-map space. Our main aim is to study the isometries of a
subclass of instanton corrected c-map spaces associated to projective special
real (PSR) manifolds with a compatible mutually local variation of BPS
structures. We call the latter subclass instanton corrected q-map spaces. In
the setting of Calabi-Yau compactifications of type IIB string theory,
instanton corrected q-map spaces are related to the hypermultiplet moduli space
metric with perturbative corrections, together with worldsheet, D(-1) and D1
instanton corrections. In the physics literature, it has been shown that the
hypermultiplet metric with such corrections must have an
$\mathrm{SL}(2,\mathbb{Z})$ acting by isometries, related to S-duality. We give
a mathematical treatment of this result, specifying under which conditions
instanton corrected q-map spaces carry an action by isometries by
$\mathrm{SL}(2,\mathbb{Z})$ or some of its subgroups. We further study the
universal isometries of instanton corrected q-map spaces, and compare them to
the universal isometries of tree-level q-map spaces. Finally, we give an
explicit example of a non-trivial instanton corrected q-map space with full
$\mathrm{SL}(2,\mathbb{Z})$ acting by isometries and admitting a quotient of
finite volume by a discrete group of isometries.
