There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become infinite loop spaces and the map above can be promoted to an infinite loop space map.

>Are there any other infinite loop space maps into or our of $Aut(F_{\infty})?$

Some notes: Galatius has shown relatively recently that $BAut(F_{\infty})^+$ is homotopic to the sphere spectrum; the latter has been known for about 4 decades to be homotopic to $BS_{\infty}^+$ (Barratt-Priddy-Segal). So loop maps in or out of the symmetric group can all be defined up to homotopy on $Aut(F_{\infty})$ instead. Also, in all of the above, can "Aut" be replaced by "Out"?