Modular distribution of invariants for topo-symmetric extensions

Let $G$ be a finite group and $H$ a discrete abelian group. Write $\mathrm{Ext}_{\mathrm{ts}}(G,H)$ for the set of topo-symmetric extensions of $G$ by $H$, and let $\dim(E)$ denote the dimension invariant of such an extension. The invariants of topo-symmetric extensions, such as dimension and density, are distributed uniformly modulo $\gcd(|G|,|H|)$:

$$\#\{ E \in \mathrm{Ext}_{\mathrm{ts}}(G,H): \dim(E) \equiv k \pmod{\gcd(|G|,|H|)} \} \sim \frac{|\mathrm{Ext}_{\mathrm{ts}}(G,H)|}{\gcd(|G|,|H|)}.$$

Modular distribution of invariants. The invariants of topo-symmetric extensions are distributed uniformly modulo $\gcd(|G|,|H|)$ in the sense above. Numerical experiments for small cyclic groups suggest that this uniformity holds for most classes of extensions, but a general proof is still open.