Events of Gibeah
First, let's recap the events of Judges 19-20:
A mysterious Levite from the remote parts of mount Ephraim goes to retrieve his concubine (a concubine is a wife without an inheritance) and take her to his house, the "House of YHWH". His concubine was born in Bethlehem and has been staying with her father for about 4 months. (Four months is roughly the time from Hannukah to Passover).
The Levite rides into the city of his concubine's father with two donkeys on the first day and is greeted with joy. On the afternoon of the fifth day, he leaves her father's house and goes to Gibeah, where in the evening he meets a countryman of his, enters his house, and has a meal with footwashing. The Levite brings the food that they both eat.
After the meal, when it is already dark, the wicked men from Gibeah surround the house demanding to rape the Levite, similar to the events of Sodom. Except this time, the Levite throws his concubine to the crowd allowing them to rape her all night.
On the morning of the sixth day, she is dead. The Levite puts her on his donkey to take her back to his house -- the house of the YHWH. In his house, he cuts her up into 12 pieces and sends each piece to a tribe of Israel, which unites then as one man (you can imagine all the pieces of the concubine being brought together) to wage a war against the Benjamites in Gibeah.
This war lasts for three battles, but only the tribe of Judah participates.
At the end of the third battle, there are 600 Benjamites clinging to the rock of Rimmon, and they are the remnant that are spared. Brides are then found for them. Peace offerings are given before the third battle, symbolic of reconciliation, but only with a remnant -- that remnant that was clinging to the rock.
Did God cause this or permit it?
This depends on your theology. If you are a polytheist or Zoroastrian and think there is some battle between God and the Devil, where sometimes the Devil wins but other times God wins, and you think God will get the final victory in some distant future despite being bested a few times, then you can say that God "permitted it" but wished it wouldn't happen.
If you are a monotheist who believes there is only one God in charge of the universe, then God is responsible for the events of Gibeah just as he is responsible for the events in Sodom, or the events in Jerusalem on passover night, which were also a terrible abuse of an innocent of which the events of Gibeah are a type. For the monotheist, all of these events fulfill God's pre-ordained will:
Woe unto the world because of offences!
for it must needs be that offences come;
but woe to that man
by whom the offence cometh!
In the monotheist view, evil is just a tool that God uses, together with good, in order to accomplish his will, and whether we are talking about Gibeah or Jerusalem, God's will is responsible for whatever happens in those cities, as it says in Amos 3.6 (KJV):
Shall a trumpet be blown in the city, and the people not be afraid?
Shall there be evil in a city, and the LORD hath not done it?
Or we read in Isaiah 45.7:
I form the light, and create darkness:
I make peace, and create evil:
I the Lord do all these things.
So we can ask, was the abuse and murder of Christ good or evil? Was the death of the firstborn in Egypt good or evil? Was the murder of the Levite's concubine good or evil? If you read the Bible with a view towards wagging your finger at things you find evil -- thus judging God -- you will not perform good exegesis, as everything will be filtered through judgmental morality of a fallen and irreconcilably depraved mind that is unable to discern God's will.
It was clearly evil, but it was part of God's will, so it was ultimately good. That is why we call it "Good Friday". Good Friday was not an example of God being "bested" by Satan, or "permitting" man to do something that God wishes didn't happen. No, God required it, God caused it to happen, God is responsible for it -- but woe to vessels of wrath that performed the deed. Just as God raised up Pharaoh and hardened his heart so that the firstborn would be killed and Israel delivered on another passover.