Abstract: The Boone-Higman conjecture (1973) predicts that a finitely generated group has solvable word problem if and only if it embeds in a finitely presented simple group. The "if" direction is true and easy, but the "only if" direction has been open for over 50 years. The conjecture is known to hold for various families of groups, perhaps most prominently the groups \(\mathrm{GL}_n(\Z)\) (due to work of Scott in 1984), and hyperbolic groups (due to work of Belk, Bleak, Matucci, and myself in 2023). In this talk I will discuss some recent work joint with Belk, Fournier-Facio, and Hyde establishing the conjecture for \(\mathrm{Aut}(F_n)\), the group of automorphisms of the free group \(F_n\). I will also highlight an interesting sufficient condition for satisfying the conjecture, which just amounts to finding an action of the group with certain properties, with no need to actually deal with simple groups.