This is more common than most people might think. The GF claims that BF did not have permission, thus (her) auto policy does not pay. By stating no permission, then insurance does not pay, nor does this accident goes on the GF record of a claim. But then the insurance would investigate on (who) is telling the truth.
Now her insurance could deny the claim for several reasons. They could deny it since the BF has no license, but with many policies unless there is an (exclusion) in the policy that they don't pay for any claim if the driver has no license, but (most) policies had (no) exclusion, thus many will pay. But since living together, then it is (expected) that the BF would/could have permission to drive the vehicle, since living together.
Now, (if) insurance denies for lack of permission, or for not having a license, then that is then considered a "denial of coverage", thus if the (other) driver has uninsured motorist property damages, then (their) policy will pay for the damages. No coverage is another word for (no) insurance.
If no collision or uninsured property damage coverage, then the other driver would have to sue the (owner-GF) or the driver-BF to be paid back.
retired auto adjuster