Solvable not implies solvable automorphism group

Statement
It is possible to have a solvable group $$G$$ such that the automorphism group of $$G$$ is not a solvable group.

Proof
Let $$G$$ be elementary abelian group:E8, which is a three-dimensional vector space over field:F2. The automorphism group is GL(3,2), which is a finite simple non-abelian group of order 168, hence is not solvable.