It certainly looks like there is new bone growth, doesn't it?

No... it's not bone growth. It, um... the lack of splintering or cracking indicates that the person was not only alive when this happened, but it was hammered through the skull (as opposed to being shoved through with a single thrust). Hopefully they were at least unconscious, though i wouldn't count on it.

or she

If the museum had done any repair work it would have been in a contrasting coloured medium so we could tell

I think it just chipping of the bone around the entry point; the difference in surface texture makes it look like it's something else.

The discolouration could due to iron oxides leaching into the bone from the metal of the spike. There are a couple of articles about this, but they don't really hold true, one of them stating that the spike exited through the mouth, which this one clearly did not. So it could be a fake or reconstruction.

Doesn’t look to me like he survived from this photo

Given the unment fracture going from the spike entry downwards and that one hole at the end of it, I suspect a lower left chunck was broken out when found and just glued back together a little too well...

It's not unheard of. Phineas Gage survived a railway spike through his head. Granted, he had suffered severe memory impairment, and by the accounts I've read, his personality did a full 180, but still, he survived.

It looks like a repair to keep the spike in place. The area immediately surrounding the spike doesn’t look quite like the rest of the skull.

I can find absolutely no credible source for this image online. It's also very recent, so there's a possibility it's computer generated. Given the observations about the condition of bone around the spike, I'm thinking this is fake.

That's what I was looking at. It certainly looks like there are no gaps in the skull around the spike. That would imply at lest some growth. One piece I read about it seems a bit contradictory, it says the spike was driven in the back of the skull and out the mouth. That doesn't appear to be the case from the photo. So I'm not sure whether it's other "facts" are right either.

Phineas Gage!