The answer to this question might be known, but I don't know of any reference and will appreciate any references.


Let $n>0$. By Snaith splitting there is a stable splitting of $\Omega^kS^{n+k}$ into wedge of spaces $D\_{k,r}S^n=F(\mathbb{R}^k,r)\ltimes\_{\Sigma\_r}(S^n)^{\wedge r}$ as


$$\Sigma^\infty\Omega^kS^{n+k}\simeq\bigvee\_{r=1}^{+\infty}\Sigma^\infty D\_{k,r}S^n$$


where $F(\mathbb{R}^k,r)$ is the configuaration space of $r$ distinct points in $\mathbb{R}^k$. 
It is known that $D\_{k,2}S^n=\Sigma^nP\_{n}^{n+k-1}$ where $P\_n^m$ is the truncated real projective space $P^m/P^{n-1}$. 


What I would like to know is about other pieces and specially $D\_{k,2^s}S^n$ with $s>0$. Is it known that for any positive $k,s,n$ the space $D\_{k,2^s}S^n$ is suspension of some $CW$-complex? If the answer is positive, do we know how many suspension may appear?



Stable summands of $\Omega^kS^{n+k}$
The full story is eluded to in the comments of Arone and Rognes. First of all, $D\_{k,r}\Sigma^n X$ is always an $n$-fold suspension. More exciting is that, for all pairs $(k,r)$, there is a natural number $d = d(k,r)$, such that
$D\_{k,r} \Sigma^dX = \Sigma^{rd}D\_{k,r}X$.


So what is $d$? It is the order of the canonical $r$-dimensional vector bundle over the configuration space $F(\mathbb R^k, r)/\Sigma\_r$. This is a bundle of finite order because it has finite structure group and a finite dimensional base space. The actual numbers $d(k,r)$ were computed a long time ago: see [Cohen, Cohen, Kuhn, Neisendorfer, Bundles over configuration spaces, Pac. J. Math. 104 (1983), 47-54]. And yes, $d(2,r)=2$ for all $r$, was known even earlier.


