The skull has definitely been proven fake, although I'm intensely skeptical of using a site called 'educate yourself' that talks about a bunch of conspiracy theories as a source.

In fact, I'm going to call bunk on a lot of what the poster said. They talked about Thomas Cadet, and the picture he took of Hitler, but... it's an unsourced photo, there's no mention of it anywhere I can find, and 'Thomas Cadet' pulls up nothing. There were no known pictures taken of Hitler after his death, and I talk a bit about why that was over here.

"The Soviets used the suicide narrative as a photo op. But, after Perostroika and the old Soviet files were released, we found out that Stalin had no idea what had happened to Hitler. Years after claiming that Hitler killed himself, Stalin himself was asking where Der Fuhrer was. (In the files, we found out that he believed that he'd escaped to either Spain or Argentina.)"

The soviet story actually makes sense when you realize that the cold war was just beginning. Their original story was that Hitler had escaped, and that the western allies (notably America) were hiding him. This led to confusion and doubt among the western allies, which was no doubt their goal.

In the meanwhile, SMERSH had collected the remains of several people. Hitler, Eva Braun, Goebbels, his wife, and their children. They had an autopsy report conducted to confirm their identities, and then they were all buried under a parking lot, of all places. The SMERSH report was actually published in America at this point, but was dismissed because it was coming out of Russia, and tensions were still fairly high.

The skull fragment comes in later, as it was recovered from the mass grave before the bodies were destroyed. All the bodies were in a very advanced state of decay, and there was no real way to determine which body was which. It's entirely possible that someone did think the skull was Hitlers--but that they were confused and grabbed the wrong skull.