Given an orbit space $M/\Gamma$ and an equivalence relation
defined in it by means of the action of a group $G$, we obtain a
miniversal deformation of an orbit through a miniversal
deformation in $M$ with regard to a suitable group action of
$G\times \Gamma$. We show some applications to the perturbations
of $m$-tuples of subspaces and $(C,A)$-invariant subspaces.