For which $n$ is there a unique perfect group of order $n$? Are there infinitely many such $n$?

Some guesses for infinite sequences of such $n$: $|\mathrm{PSL}(2,p)|$, $|\mathrm{SL}(2,p)|$, $|A_m|$, $|S_m|$.

If you replace "perfect" with "simple", a complete understanding follows from the Classification. Apart from a few small coincidences and one well-understood infinite family, there are no coincidences. See this previous question for instance. But the picture for perfect groups is less clear to me.

A complete list of the orders of perfect groups up to $10^6$ is contained in Section 5.4 of the book by Holt and Plesken (*Holt, Derek F.; Plesken, Wilhelm*, Perfect groups. (1989). ZBL0691.20001.). There seem to be many $n$ for which there is a unique perfect group of order $n$, but there are also many $n$ (e.g. $2^{14} \cdot 60$) for which the number of perfect groups is very large. There are precisely five $n < 10^6$ for which there are both simple and non-simple perfect $G$ of order $n$:

- 20160: $A_8$, $L_3(4)$, $2.(A_5 \times L_3(2))$
- 181440: $A_9$, $A_6 \times L_2(8)$, $L_3(2) \times 3.A_6$
- 262080: $L_2(64)$, $2^2.(A_5 \times L_2(13))$
- 443520: $M_{22}$, $2^2.(L_3(2)\times L_2(11))$
- 604800: $J_2$, $A_5^2 \times L_3(2)$, $2^2.(A_5 \times A_7)$