Weak heirs, coheirs and the Ellis semigroups

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 $S_{ext,G}(N)$.