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Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?

Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$. On the other hand $H^2$...