header image on front page only

Have you set up a Child Theme?

Yes.

So presuming you’ve copied the Twenty Twelve header.php file into your Child Theme, edit that header.php file and replace all of its code with the code from this PasteBin page: https://pastebin.com/kEZ8mvL2 [Note: I’ve just updated the PasteBin]

Does that do what you want?

Ah, nearly. The first version you posted didn’t work at all, the second doesn’t display the header on any page except the Blog page. I have a static front page though which is not the Blog page and it does not show there. Is that something to do with the difference between “home” page and “front” page?

I think so – try this pastebin instead: https://pastebin.com/Df9BJ7ze

Perfect! Thanks for your help ??