Jean-Luc Loday told me about the extended action by $\Sigma_{j+1}$ in the fall of 1992, after an Oberwolfach talk I gave about the rank filtration of algebraic $K$-theory, where the $\Sigma_j$-representations given by the integral homology of the Goodwillie derivative  spectra $W_j$ played a role. I had shown that these representations were freely generated by $j$-fold Lie brackets, and Loday knew the connection to spaces of trees. Maybe you were there, too?