It is well known, e.g. by Cohen's "A model for the free loop space of a suspension", that there is a stable splitting of the free loop space $\mathcal{L} \Sigma X $of the suspension $\Sigma X $ given by
$$ \Sigma^{\infty} \mathcal{L} X \simeq \bigoplus_{n>1} \Sigma^{\infty}(S^1_+ \wedge_{C_n} X^n) $$
where $C_n$ is the cyclic group of order $n$, and $S^1_+ \wedge_{C_n} X^n$ is the $S^1$-space induced by the inclusion of groups $C_n \to S^1$ and viewing $X^n$ as a $C_n$-space.
I would like to know if this can be promoted to an $S^1$-equivariant splitting, at least in the case when $X$ is an $k$-sphere. There seems to be some equivariance going on, as the map is induced by a map (at the level of spaces)
$$ X^n \to \mathcal{L}X $$
which is equivariant with respect to the $C_n$-action on the left, and the $C_n$-action on the right obtained by restricting the $S^1$ action. Any input would be greatly appreciated!
