Finite groups with large Chebotarev invariant

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ is said to invariably generate $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The authors recently showed that for each $\epsilon>0$, there exists a constant $c_{\epsilon}$ such that $C(G)\le (1+\epsilon)\sqrt{|G|}+c_{\epsilon}$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each $\alpha>0$ there exists an absolute constant $\delta_{\alpha}$ such that if $G$ is a finite group and $C(G)>\alpha\sqrt{|G|}$, then $G$ has a section $X/Y$ such that $|X/Y|\geq \delta_{\alpha}\sqrt{|G|}$, and $X/Y\cong \mathbb{F}_q\rtimes H$ for some prime power $q$, with $H\le \mathbb{F}_q^{\times}$.