Minimal subdynamics and minimal flows without characteristic measures

Given a countable group G and a G-flow X, a probability measure $\mu $ on X is called characteristic if it is $\mathrm {Aut}(X, G)$ -invariant. Frisch and Tamuz asked about the existence of a minimal G-flow, for any group G, which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups $\{\Delta _i: i\in I\}$ , when is there a faithful G-flow for which every $\Delta _i$ acts minimally?