This is a very interesting question.
What I believe is in the character of God and we are to seek understanding of the scriptures. Not just Greek/Hebrew meanings but look in the overall context of the bible. The bible usually interprets itself.
What does the bible say about the character of God?
If ye then, being evil, know how to give good gifts
unto your children, how much more shall your Father
which is in heaven give good things to them that ask
Do not be deceived, my dear friends! Every good gift and every perfect present comes from heaven; it comes down from God.
So, if I look into the context of other scriptures, God will not send evil gifts but will only send good gifts. How do we then explain the evil spirit from the Lord?
Let's look at the context of the story of Job. Job, being a godly man, has yielded himself to God. God was in control of his life. When the devil wanted to attack Job, he had to go through God first and that he did. God permitted the devil to touch Job.
I believe that Saul was in a similar situation. God protected Saul and His Holy Spirit was upon Saul. When Saul disobeyed and chose to rebel, God was still in control and He allowed satan to take back control of Saul. It was like what the guest asked, God gave him up.
Whose fault was it? Was it God's fault when He gives us up to our desires. We have a free will and if we decide to live in bitterness, jealousy and rebellion, we actually give ourselves up again to evil spirits. However, when we are believers, we believe that our lives are still in God's control. He is sovereign and whatever happens to us is permitted by God. Do we then say that God sent evil into our lives? That all boils down again to the concept of free will. We can choose in life because we have free will, but we cannot choose the consequences. There is still cause and consequences in life. These laws are still in operation. It's only by the supernatural grace of God that sometimes we escape the consequences. Even David have to live with the consequences of his sin when he went with Bathsheeba. God may allow us to suffer the consequences, but His grace is always there and He is still in control. If we continue to yield ourselves to Him, all things can work together for good.
I believe that Saul also had that choice. He could have yielded to God even when he lost his "king"ship. But he chose not to and gave himself over to the evil spirit with the permission of God.
There's a very good article on this: