$H=[48,28]$ or $H=[48,29]$. Also, $\delta^\natural_3(G) = \frac 27$ if and only if $G$ is an extension of $\operatorname{PSL}(2,7)$.

Here $G$ being an extension of $H$ by some $N \lhd G$ means $G/N \simeq H$. The groups $[48,28]$ and $[48,29]$ are the two groups of order 48 which are extensions of $S_4$ by the cyclic group $C_2$ and contain $\operatorname{SL}(2,3)$.

The $n=1$ case is already contained in Proposition \[prop1\] as $d_1 = d_1^\natural$. For $n=2, 3$, these bounds are much better than the trivial bounds $d_2^\natural \ge \frac 18$ and $d_3^\natural \ge \frac 1{18}$ from Proposition \[prop1\]. For $n=2$, related results were previously obtained by the second author in [@walji] and will be discussed below.

Note that while $d_n$ is a strictly decreasing sequence for $n \ge 1$, our result says this is not the case for $d_n^\natural$.

In a slightly different direction, one can look for stronger lower bounds than $\frac 1{2n^2}$ for certain families of groups. We do not begin a serious investigation of this here, but just treat two basic families of finite groups of Lie type which are related to the calculations for $\delta^\natural_2(G)$ and $\delta^\natural_3(G)$.

\[thm2\] We compute $\delta_n(G)$ and $\delta^{\natural}_n(G)$ where $G = \operatorname{PSL}(2,q)$ and 