A finite Q-bad space

We prove that for a free noncyclic group $F$, $H_2(\hat F_\mathbb Q, \mathbb Q)$isanuncountable$\mathbb Q$-vectorspace\mathrm{.}Here$\hat F_\mathbb Q$isthe$\mathbb Q$-completionof$F$. This answers a problem of A.K. Bousfield for the case of rational coefficients. As a direct consequence of this result it follows that, a wedge of circles is $\mathbb{Q}$-bad in the sense of Bousfield-Kan. The same methods as used in the proof of the above results allow to show that, the homology $H_2({\widehat{F}}_{\mathbb{Z}},\mathbb{Z})$ is not divisible group, where ${\widehat{F}}_{\mathbb{Z}}$ is the integral pronilpotent completion of $F$.