Minimal subdynamics and minimal flows without characteristic measures

Given a countable group \(G\) and a \(G\)-flow \(X\), a measure \(\mu\in P(X)\) 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?