Assume $G\prec H$ are groups and ${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$. We apply these results in the model theoretic context. Namely, assume G is a group definable in a model M and $M\prec ^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups $S_{ext,G}(M)$ and $S_{ext,G}(N)$. Assuming every minimal left ideal in $S_{ext,G}(N)$ is a group we prove that the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$.

Assume G is a group definable in a model M of a stable theory T. We prove that the semigroup SG (M) of complete G-types over M is an inverse limit of some semigroups type-definable in Meq. We prove that the maximal subgroups of SG (M) are inverse limits of some definable quotients of subgroups of G. We consider the powers of types in the semigroup SG (M) and prove that in a way every type in SG (M) is profinitely many steps away from a type in a subgroup of SG (M).