Solvable automorphism group implies solvable of derived length at most one more

Statement
Suppose $$G$$ is an group whose automorphism group is solvable, i.e., a group whose automorphism group is solvable. Then, $$G$$ is also solvable. Moreover, if the derived length of the automorphism group of $$G$$ is $$l$$, the derived length of $$G$$ is at most $$l + 1$$.

Similar facts

 * Aut-abelian implies class two
 * Aut-nilpotent implies nilpotent of class at most one more
 * Trivial automorphism group implies trivial or order two

Converse
The converse is not true: solvable not implies aut-solvable.

Facts used

 * 1) uses::Group acts as automorphisms by conjugation
 * 2) uses::Solvability is subgroup-closed
 * 3) The center of a group is abelian, hence solvable
 * 4) uses::Solvability is extension-closed