  Let $M$ be a compact and connected smooth manifold endowed with a smooth
action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse
function on $M$. We prove that the space of $\Gamma$-invariant Riemannian
metrics on $M$ contains a residual subset ${\mathcal Met}_f$ with the following
property. Let $g\in{\mathcal Met}_f$ and let $\nabla^gf$ be the gradient vector
field of $f$ with respect to $g$. For any diffeomorphism $\phi$ of $M$
preserving $\nabla^gf$ there exists some real number $t$ and some
$\gamma\in\Gamma$ such that for every $x\in M$ we have
$\phi(x)=\gamma\,\Phi_t^g(x)$, where $\Phi_t^g$ is the time-$t$ flow of the
vector field $\nabla^gf$.
