Algebra in superextensions of twinic groups

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $\lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $\beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $\lambda(X)$ in the semigroup of all self-maps of the power-set of $X$ and using this representation describe the structure of minimal ideal and minimal left ideals of $\lambda(X)$ for each twinic group $X$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group with two generators.