Solvability of the (outer) automorphism group of a central product

Let $$G = A * B$$ be a central product of the finite groups $$A$$ and $$B$$. Suppose that $$\operatorname{Aut}(A)$$ and $$\operatorname{Aut}(B)$$ are solvable groups. Then is $$\operatorname{Aut}(G)$$ a solvable group?

• Where does $G$ come into play? – the_fox Nov 8 '18 at 14:16
In general no. A counterexample with the direct product to both questions is with $$A$$ and $$B$$ both cyclic of order $$5$$. $${\rm Aut}(A \times B) = {\rm GL}(2,5)$$ is not solvable.
A counterexample with a genuine central product is $$A=D_8$$, $$B=Q_8$$, where again the outer automorphism group has $$A_5$$ as a composition factor.