Maxap groups with zero-dimensional group compactifications

Question

Suppose $G$ is a countable group. We say that $G$ is maxap if there is an injective homomorphism $\phi: G\to K$ with $K$ a compact group. My question is what we can demand of $K$. Given $G$ a maxap group, can we demand that $K$ be zero-dimensional? Certainly this is true when $G$ is residually finite but in general it seems tricky.

Answer 1

Precisely, this is possible iff $G$ is residually finite (because 0-dimensional compact groups are profinite, hence residually finite).

So $\mathbf{Q}$ is an example of maximally almost periodic with no injective homomorphism into such a $K$.

(For finitely generated groups however, maxap $\Leftrightarrow$ residually finite.)