The Denmark Conspiracy Continues! But now I offer a sort of solution...
After a few moments of trying, I came up with the solution. There is something wrong with the name of Denmark. I don't know why AGM loaths it - but that is the issue.
If I use Danmark, instead of Denmark, it works at once. E.g. Image:MiniDanmarkFlag.jpg will show the image everywhere, as opposed to the sporadic, if at all nature of replacing Denmark with Danmark in the earlier mentioned image link.
So, that is the problem. But why is my next question. I don't want to spend forever changing an "e" to an "a" - especially when the a is incorrect - so can anything be done to help?
Just to check - can you all see this image? -> Template:Country data Danmark
