Necessary and sufficient condition for a subset of $\mathbb{N}_2$ to be a base set

Question

This is a follow-up to my previous question several years ago, here: Can a number be normal in an arbitrary set of bases?. Consider $\mathbb{N}_2$, the set of natural numbers greater than or equal to $2$. I define a subset $S$ of $\mathbb{N}_2$ to be a "possible base set", if there exists a real number $r$ which is normal to every base $b$ in $S$, and not normal to every base $b$ in the complement of $S$ with respect to $\mathbb{N}_2$. My question is, what is a useful necessary and sufficient condition for a subset of $\mathbb{N}_2$ to be a possible base set? I know that not every subset of $\mathbb{N}_2$ is a possible base set.