Given a finite group $G$, define the embedding number $\phi(G)$ to be the minimum natural number $N$ such that $G$ embeds in $S_N$.
For a fixed cardinality $n$, write $\widetilde{E}(n)$ for the average embedding number of $G$ as $G$ ranges over isomorphism classes of groups of order $n$, and write $E(n)$ for the slightly more natural $$ \sum_{\#G = n} \frac{\phi(G)}{\#\text{Aut}(G)} \bigg/\sum_{\#G = n} \frac{1}{\#\text{Aut}(G)} . $$
Is anything known about the asymptotics of $E(n)$ or $\widetilde{E}(n)$? What if we restrict $n$ to ranging over powers of a fixed prime (or just $2$)?