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?
And what about the same question with outer automorphism groups instead?
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.