  Q-map spaces form an important class of quaternionic K\"ahler manifolds of
negative scalar curvature. Their one-loop deformations are always inhomogeneous
and have been used to construct cohomogeneity one quaternionic K\"ahler
manifolds as deformations of homogeneous spaces. Here we study the group of
isometries in the deformed case. Our main result is the statement that it
always contains a semidirect product of a group of affine transformations of
$\mathbb{R}^{n-1}$ with a Heisenberg group of dimension $2n+1$ for a q-map
space of dimension $4n$. The affine group and its action on the normal
Heisenberg factor in the semidirect product depend on the cubic affine
hypersurface which encodes the q-map space.
