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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 a 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.

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    $\begingroup$ You'll have no control, even on the homotopy type, if you don't assume path-connected, and you'll need even more (otherwise one can crook things by adding fake loops that are not paths). So "locally path-connected" would help too. Maybe a starting point if you want a moduli space is to figure out what you need when $G=1$. $\endgroup$
    – YCor
    Commented Nov 6, 2021 at 13:33
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    $\begingroup$ Such a moduli space, if reasonably defined, would be a union of $B(\operatorname{haut}(X))$ over all $X$ with $\pi_1(X)=G$. Here $B$ denotes the classifying space of a monoid and $\operatorname{haut}$ denotes homotopy automorphisms. Spaces like this are useful, but not to study $\pi_1$. $\endgroup$
    – Connor Malin
    Commented Nov 6, 2021 at 14:29
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    $\begingroup$ Better put a bound on the number of cells in $X$, or demand that the homotopy groups of $X$ are finitely generated, or something else along those lines. Otherwise you have a proper class of such spaces $X$ (this already happens when $n=2$ and $G$ is trivial), so no hope of such a moduli space. $\endgroup$
    – user164898
    Commented Nov 6, 2021 at 15:15

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