  We introduce a notion of formally \'etale $\mathbb{E}_{\infty}$-coalgebras
and show that they admit essentially unique, functorial lifts along square zero
extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a
perfect $\mathbb{F}_p$-algebra $k$, Weil restriction along the augmentation
$\mathbb{W}(k)\to k$ induces a fully faithful functor from formally \'etale,
connective $\mathbb{E}_{\infty}$-coalgebras in $k$-modules to connective
$\mathbb{E}_{\infty}$-coalgebras in $p$-complete modules over the spherical
Witt vectors $\mathbb{W}(k)$. Finally, we prove that for any connected space
$X$, the $k$-homology $k[X]$ is a formally \'etale
$\mathbb{E}_\infty$-coalgebra in $k$-modules. This shows that
$\mathbb{W}(k)[X]^{\wedge}_p$ can be recovered as the essentially unique lift
of $k[X]$ to a connective coalgebra in $p$-complete $\mathbb{W}(k)$-modules.
