In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation group are considered and central limit theorems for kind of balls volumes are dicussed. Some of these theorems goes back to classical results in statistics by Kendall, Mann-Whitney etc.. and some completely new.

Question in brief: Are there any generalizations of such results and constructions to other finite groups ? One may expect kind of classes/examples of pairs: (group, metric on group) such that kind of central limit theorems for balls volumes holds true.

Let me give more details:

Simple Motivational Example: Consider finite abelian group $(Z/2Z)^n$ i.e. just sequences of zeros and ones. We can put metric on it: $d(x,y)=|x-y|_{L_1} = \sum_i |x_i-y_i|$ i.e. just count all "1" in vector $x-y$. A volume of ball of radius "k", is the same as probability of sum $\sum_{i<=n} \xi_i$ to be less or equal than $k$ (and multiply by $2^n$), where independent variables $\xi_i = 0,1$ with $1/2$-probability. That is exactly the setup of the most classical central limit theorem.

Diaconis-Graham theorem 1977
Consider symmetric group $S_n$ with a metric $\rho(\pi,\sigma) = \sum |\pi(i)-\sigma(i) |$. If $\sigma$ is chosen uniformly in $S_n$ then:

$$ P( \frac{\rho - Average}{\sqrt{Variance}} < t )= 1/sqrt{(2\pi)}\int_{\infty}^t exp(-x^2/2) dx + O(1)$$

$$ Average(\rho) = (n^2-1)/3, Variance(\rho) = 1/45(n+1)(2n^2+7). $$

Diaconis also describes similar results for other of metrics on symmetric group. Some of them goes to classical results in statistics. And related to rank correlation coefficients by Kendall and Spearman, to Mann-Whitney test, etc (see also MO)... See also recent paper A central limit theorem for a new statistic on permutations Sourav Chatterjee, Persi Diaconis where other generalization for symmetric groups is considered. It is tempting to think, that similar results can be generalized substituting symmetric group by some other groups.