In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

Originally, this question asked if there's an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{S}_*,\wedge,S^0)$, for $1\leq k\leq\infty$.

At least in its most basic sense (i.e. that $\mathbb{E}_{k}$-comonoids in $\mathcal{S}_*$ are just pointed spaces), there isn't, as pointed by Denis Nardin here.

• Is there some more refined analogue of Péroux–Shipley's result in this context? That is, is there some sense in which $\mathbb{E}_{k}$-comonoids in $\mathcal{S}_*$ are "not too far" from being just pointed spaces?
• What are some natural examples of such objects?