  We study symmetry properties of quaternionic K\"ahler manifolds obtained by
the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal
automorphisms of the initial hyper-K\"ahler data we associate a central
extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the
resulting quaternionic K\"ahler manifold. More specifically, we study the
metrics obtained by the one-loop deformation of the $c$-map construction,
proving that the Lie algebra of infinitesimal automorphisms of the initial
projective special K\"ahler manifold gives rise to a Lie algebra of Killing
fields of the corresponding one-loop deformed $c$-map space. As an application,
we show that this construction increases the cohomogeneity of the automorphism
groups by at most one. In particular, if the initial manifold is homogeneous
then the one-loop deformed metric is of cohomogeneity at most one. As an
example, we consider the one-loop deformation of the symmetric quaternionic
K\"ahler metric on $SU(n,2)/S(U(n)\times U(2))$, which we prove is of
cohomogeneity exactly one. This family generalizes the so-called universal
hypermultiplet ($n=1$), for which we determine the full isometry group.
