Random pro-p groups via iterated uniformly random central extensions

Question

Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:

We want to construct an inverse system $$\cdots \xrightarrow{\alpha_i} G_i \xrightarrow{\alpha_{i-1}} \cdots \xrightarrow{\alpha_2} G_2 \xrightarrow{\alpha_1} G_1$$ where $G_i$ is a group of order $p^n$ and $\alpha_i$ fits in a central extension $$ 1 \rightarrow \mathbb F_p \rightarrow G_{i+1} \xrightarrow{\alpha_i} G_i \rightarrow 1$$

We start with $G_1 = \mathbb F_p$ and if we constructed the inverse system up to $G_i$, we take an uniformly random element of $\eta \in H^2(G_i,\mathbb F_p)$ and choose a $G_{i+1} \xrightarrow{\alpha_i} G_i$ which fits in a central extension corresponding to $\eta$.

Then we can take the limit $\hat G$ of this inverse system. This should define a probability measure on the isomorphism classes of pro-$p$ groups of cardinality $2^{\aleph_0}$.

This description is maybe a bit sloppy concerning set theoretical issues, so to make it work in any case, we could refine us to a skeleton $\mathcal C$ of the category of p-groups (with $\mathbb F_p \in \mathcal C$) and choose maps $$ H^2(G,\mathbb F_p) \rightarrow \{1 \rightarrow \mathbb F_p \rightarrow E \xrightarrow{\alpha} G \rightarrow 1 \text{ central extension in } \mathcal C\}$$ in beforehand.

Questions

1. I would expect $\hat G$ to be "as bad as possible". E.g. it should be almost surely non-abelian. Is it almost surely topologically finitely generated? Is it also almost surely not solvable?
2. Given a finite p-group $H$, is it almost surely a quotient of $\hat G$? This would answer the first question immediately.
3. Is there an interesting property of $\hat G$ which has probability $0 \lt q \lt 1$?
4. As a negative answer to question 3, is there a pro-$p$ group $\hat H$, so that $\hat G \cong \hat H$ almost surely? If so, can you give a nice description of $\hat H$? For example, is it the free pro-$p$ group on countable many generators?

Edit: I am not so sure anymore whether it's almost surely not topologically finitely generated. $\hat G$ is finitely generated if $\lim_{n \to \infty} d(G_n) \lt \infty$ where $d(G_n)$ is the minimal number of generators. I think that $d(G_{n+1}) = d(G_n) + 1$ holds for $\eta = 0 \in H^2(G_i, \mathbb F_p)$ and $d(G_{n+1}) = d(G_n)$ for all nonzero $\eta \in H^2(G_i, \mathbb F_p)$. By Golod-Shafarevich $$\dim H^2(G_n, \mathbb F_p) \gt \frac{1}{4}\cdot d(G_n)^2$$ so $d(G_n)$ will grow very slowly. But this is of course not enough to show that $\hat G$ is almost surely finitely generated.

Answer 1

This is only a partial answer regarding 1. For a $p$ group of cardinality $p^n$, we can consider its nilpotency class $c$ and its coclass $r = n - c$. If $\eta = 0 \in H^2(G_n, \mathbb{F}_p)$, then the nilpotency class of $G_{n+1}$ will be the same as the one of $G_n$, so the coclass increases by one.

Now we can use the following facts: Given $p$ and $r$, there are only finitely many cohomology rings $H^*(G, \mathbb{F}_p)$ of finite $p$-groups $G$ wth coclass $r$. For $p=2$, this is due to Jon F. Carlson, for odd $p$, I found a result by Peter Symonds (arXiv link). With this in hand we can show that the coclass of $G_n$ goes to infinity almost surely: otherwise, the dimension of $H^2(G_n, \mathbb{F}_p)$ stays bounded and we almost surely hit $\eta = 0$.

Similarly, I think one can apply the result of this paper to show that almost surely the rank or the nipotency class of $G_n$ goes to infinity.