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