As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^{\infty} \dashv \Omega^{\infty}$ adjunction.

Is it also the case that the category of spaces is comonadic over the category of spectra (or maybe connective spectra) ?

If so do we have a nice description of "what is a (unstable) space" in terms of stable homotopy theory, i.e. can we say something about what are the $\Sigma^{\infty} \Omega^{\infty}$-coalgebra in spectra in terms of more familiar structure (in the same way that a $\Omega^{\infty} \Sigma^{\infty}$-algebra is the same as a group-like $E_{\infty}$-algebra).

For example $\Sigma^{\infty} X$ naturally has the structure of a cocomutative (in the $E_{\infty}$-sense) co-algebras (coming from the diagonal map of $X$). Can we characterize $\Sigma^{\infty} \Omega^{\infty}$-coalgebra, as such cocommutative coalgebra satisfying some additional conditions ?

I'm interested in how to deduce some properties of a higher toposes from properties of its category of spectra, and more generally, how much of a higher topos can be understood from its category spectra. so this sort of explicit description could be useful and any description of the co-monad only involving more classical structure on the category of spectra (like the smash product and the subcategory of connective spectra) might be useful.