Projective objects in the category of finite groups 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).

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 $H^2(G,M)\
eq0$. 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 $H^k(G,\mathbb{F}_p)\
eq0$ 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.