Imaginaries in pairs of algebraically closed fields

We consider the theory ( A C F p ) P of pairs F < K of algebraically closed fields of a given characteristic p . We exhibit a collection of additional sorts in which this theory has geometric elimination of imaginaries. The sorts are essentially of the form ∪ a ∈ B ( F ) V a ( K ) / G a ( F ) , where G , V , B are varieties over the prime field, G is a group scheme over B and V is a scheme over B (both with irreducible fibres), G acts algebraically on V over B , and for generic b ∈ B ( F ) the action of G b ( F ) on V b ( K ) is generically free (namely regular on generic orbits).