at.algebraic topology – Is there a “spectral exterior algebra” construction in higher algebra?

Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $mathrm{Sym}_R(E)$ of $E$ over $R$, defined by
$$
begin{align*}
mathrm{Sym}_R(E) &overset{mathrm{def}}{=} mathrm{colim}_{mathbb{F}}(Delta_{E})\
&cong bigoplus_{ninmathbb{N}}E^{otimes_mathbb{S}n}_{mathsf{h}Sigma_{n}},
end{align*}
$$

where $mathbb{F}overset{mathrm{def}}{=}mathsf{FinSets}^simeq$ is the groupoid of finite sets and permutations. As A Rock and a Hard Place showed here, the $mathbb{E}_infty$-ring $mathrm{Sym}_R(E)$ comes with a natural grading by the sphere spectrum, inducing a $mathbb{Z}$-grading on $pi_0(mathrm{Sym}_R(E))congmathrm{Sym}_{pi_0(R)}(pi_0(E))$. So e.g. picking $R=E=mathbb{S}$, gives
$$
begin{align*}
pi_0(mathrm{Sym}_{mathbb{S}}(mathbb{S})) &overset{mathrm{def}}{=} pi_0(mathbb{S}{t})\
&cong mathbb{Z}(t),
end{align*}
$$

which carries the natural $mathbb{Z}$-grading.

However, the $pi_0$ of an $mathbb{S}$-graded ring can be more complicated than just a commutative $mathbb{Z}$-grading, and for instance allows for the multiplication on the $pi_0$ to be supercommutative, satisfying $ab=(-1)^{deg(a)deg(b)}ba$. This led me to the following pair of questions:

  1. Is there an “spectral exterior algebra” construction $bigwedge_RE$, whose $pi_0$ is the $mathbb{Z}$-graded supercommutative exterior algebra $bigwedge_{pi_0(R)}pi_0(E)$? If so, does it come with an $mathbb{S}$-grading?
  2. One of the more homotopy-theoretic points of view on symmetric and exterior algebras is that the passage from the former to the latter corresponds to considering a larger portion of the sphere spectrum. More generally, do we have an $mathbb{N}$-indexed sequence of “higher exterior algebra” constructions $mathrm{Sym}_R(E)$, $bigwedge_R(E)$, $bigwedge^{mathbf{2}}_R(E)$, $ldots$?