In (Brown, Cohomology of groups, p. 208, 4 (a)) the following version of the universal coefficient theorem is stated: if $G$ is a group of type $FP_\infty$ and $F$ is a flat $\mathbb{Z}[G]$-module, then for all $n \geq 1$ the canonical map $H^n(G, \mathbb{Z}) \otimes F \to H^n(G, F)$ is an isomorphism of abelian groups.
My question is, if $G$ is of type $FP_n$, then does this hold in degree $n$? In particular, if $G$ is finitely presented, does it hold in degree $2$? I did not find this anywhere in Brown's book.
I would be happy even with the case in which the action of $G$ on $F$ is trivial.