Here's a hint: We have no adequate test for determining that. The only studies we have are of purely matrilineal mitochondrial DNA.
Mitonchondrial DNA, in other words, says nothing about who your daddy was. There is no sexual recombination at all.
Why is that a problem? Some few hundreds of neanderthals here and there throughout Europe mate with those pesky new gracile, high-brow types swarming into the region. They aren't a big part of the mix, but they merge in.
Let's pick a male. He marries a Cro-magnon female. His kids get half his nuclear genes. How much of his mtDNA do they have? Not one of them has any.
Let's pick a female neandethal who marries a Cro-magnon male. Again, her kids each get half of their nuclear DNA from her. How much of their mtDNA do they have? All of them have all of it. "Now we're cooking!" you say.
But let's be real and say half of her brood is male. None of her granchildren from them will have neanderthal mtDNA. It's only her direct female line that will carry the mtDNA. She's only one woman in a village with maybe ten women in it, and the Cro-magnons think she's butt-ugly.
Chances are her mtDNA line will die out, even if her nuclear genes are spreading all over the place.