Nullstellensatz for relative existentially closed groups

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 Theorem G of [J. Algebra, 219 (1999) 16–79]. 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 qω-compact, they are geometrically equivalent.