We prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring Z[B], then the theory of B determines the one of the two-sorted structure $\langle G, B, *\rangle$ , where * is the action of B on G. More generally, we show a similar principle for structures $\langle G, B, (...) *\rangle$ , where G is a torsion-free divisible module over the quotient of Z[B] by the annulator of G. Two applications come immediately from this result: First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication. The second application is to pure groups. The semi-direct product of G by B is bi- interpretable with our structure $\langle G, B, *\rangle$ . Thus, we obtain stable decidable groups that are not linear over a field. (shrink)