Cobordism, spin structures, and profinite completions

Let $M$ and $N$ be smooth closed connected aspherical manifolds with good (in the sense of Serre) fundamental groups $G$ and $H$. We show that if $\widehat{G}\cong \widehat{H}$, then $M$ and $N$ are cobordant and the signatures of $M$ and $N$ agree modulo $8$. Moreover, $M$ is spin (resp.spin${}^{\text{CC}}$) if and only if $N$ is spin (resp.spin${}^{\text{CC}}$). We consider some analogous results for compact connected aspherical manifolds.