Projective objects in the category of finite groups

Question

It's clear $1$ is a projective object in the category of finite groups. Are there any others?

Note that the dual problem, for injective objects, is comparatively easy (using Cayley's theorem).

Answer 1

There are no others. Here's a proof using well-known facts about the cohomology of finite groups.

Let $G$ be a nontrivial finite group, and let $p$ be a prime dividing the order of $G$. Then there is a finite dimensional (and hence finite) $\mathbb{F}_pG$-module $M$ with $H^2(G,M)\neq0$. A nonzero cohomology class represents a non-split extension $1\to M\to\tilde{G}\to G\to1$.

If the existence of $M$ is not familiar, but you believe that $H^k(G,\mathbb{F}_p)\neq0$ for some $k>0$, then you can produce $M$ from the trivial $\mathbb{F}_pG$-module $\mathbb{F}_p$ by dimension shifting. Or for a specific $M$, take an exact sequence $$0\to M\to P_1\to P_0\to\mathbb{F}_p\to0,$$ where $P_1\to P_0\to\mathbb{F}_p\to0$ is a projective presentation of the trivial $\mathbb{F}_pG$-module.