Let $O$ be an augmented operad. Then there is a functor $J^O: Top_\ast \to Top_\ast$ which takes $X$ to the free $O$-algebra on $X$ subject to the condition that the nullary operation coincides with the basepoint. The Snaith splitting tells us that $\Sigma^\infty J^O(X) = \Sigma^\infty \vee_{n \geq 1} O(n)_+ \wedge_{\Sigma_n} X^{\wedge n}$. Is there a version of this where $X$ is a pointed spectrum rather than a pointed space?
That is, let $\mathbb S \to X$ be a spectrum equipped with a map from the sphere spectrum. If $O$ is an augmented operad, there should be an $O$-algebra $J^O(X)$ such that the data of a map of $O$-algebra spectra $J^O(X) \to A$ is equivalent to a map of spectra $X \to A$ commuting with the maps from $\mathbb S$.
Question: Is there an easy formula for $J^O(X)$ when $X$ is a spectrum equipped with a map $\mathbb S \to X$? Can one at least say when $J^O(X)$ is nonzero?
Apparently such a formula will have to be a little more complicated than when $X$ is a space -- in particular, the dependence on the choice of map $\mathbb S \to X$ will be more subtle. For instance, if $O$ is an $E_n$ operad and $\mathbb S \to X$ is the zero map (or more generally smash-nilpotent), then $J^O(X) = 0$, rather than some large sum of wedges of $X$.
I think there's at least a colimit formula analogous to the James construction (which is, after all, the case $O = E_1$). For instance, when $O = E_\infty$, it should be the case that $J^O(X)$ is a certain colimit indexed over the category of finite sets and injections sending $n \mapsto X^{\wedge n}$. However, what I'd really like is a formula which (like the formula of the Snaith splitting) allows one to easily see that $J^O(X) \neq 0$ in reasonable cases, and so far this colimit formula has been a little too complicated for me to say this.
