@Gillyweeds The polynomial identity $\Delta^{p-1} = 1 + \sigma + \cdots + \sigma^{p-1}$ uses the fact that $G$ has exponent $p$.

I think my previous comment is only correct if $G$ actually elementary abelian.

Sorry, I missed the condition that $G$ is a $p$-group.

The case in which $G$ is an abelian $p$-group is already not quite trivial, and it was studied implicitly by Hall in the last section of doi.org/10.1017/S0305004100031662. In this case the indices are $|G|p, |G|, |G|, \dots, |G|$. If $G$ is not a $p$-group then $W(G)$ need not be nilpotent, e.g., $C_3 \wr C_2 \cong C_3 \times S_3$ (but your question is still reasonable).

One useful fact is that the number of $N$-conjugacy classes of $p'$-elements in $xN$ is the same as the number of $x$-invariant conjugacy classes of $p'$-elements in $N$. See arxiv.org/pdf/0902.2238.pdf, Lemma 2.2.

Certainly $b$ and $c$ would have to depend on the distribution of $X$ (not just on $\mu$). This follows from taking $X_1$ to be $\epsilon^{-1}$ with probability $\epsilon$ and zero otherwise. Then $\mu = 1$ and the expectation of $\tau_1$ is $\epsilon^{-1}$.