Is the sequence $1\rightarrow[\widehat{F_2},\widehat{F_2}]\rightarrow \widehat{F_2}\rightarrow\widehat{\mathbb{Z}}^2\rightarrow 1$ split?

Question

Let $F_2$ be the free group of rank 2, and $\widehat{F_2}$ its profinite completion. Abelianization gives an exact sequence $$1\rightarrow[\widehat{F_2},\widehat{F_2}]\rightarrow \widehat{F_2}\rightarrow\widehat{\mathbb{Z}}^2\rightarrow 1$$

Surely this can't be split right? Is there a reason why it can't be? (Maybe some kind of group cohomology thing?)