The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor $$ X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]}) $$ in which $W_j$ is a spectrum with (naive) $\Sigma_j$-action, $X^{[j]}$ is the $j$-fold smash power and ${}_{h\Sigma}$ denotes homotopy orbits. The underlying unequivariant homotopy type of the coefficient spectrum $W_j$ is that of a wedge of $(j-1)!$ spheres of dimension $S^{1-j}$. The restriction of the action to the subgroup $\Sigma_{j-1} \subset \Sigma_j$ is known to be given by the permutation action.

I have been told by several folks in the past that the $\Sigma_j$-action actually extends to an action by $\Sigma_{j+1}$. The way this is somehow done, if I recall, is by considering a certain space of metric trees with $j+1$ leaves.

Does anyone know of such a construction? Does anyone know a reference?