  We prove that in every variety of $G$-groups, every $G$-existentially closed
element satisfies nullstellensatz for finite consistent systems of equations.
This will generalize {\bf Theorem G} of \cite{BMR1}. As a result we see that
every pair of $G$-existentially closed elements in an arbitrary variety of
$G$-groups generate the same quasi-variety and if both of them are
$q_{\omega}$-compact, they are geometrically equivalent.
