If we fix a group $G$ and a dimension $n$, we can ask which $n$-dimensional locally path-connected$^*$ (or otherwise sufficiently nice) topological spaces $X$ have $\pi_1(X) \cong G$. Would these spaces naturally vary in families, so we can construct moduli space $\mathcal{M}(G,n)$ of their homeomorphism classes or of their homotopy types? If the answer is yes, would the connected components of these moduli spaces naturally correspond to distinct sets of values of higher homotopy groups?

(*) Modified after YCor's comment below