Since GPL licenses may contain the notice that derivative software can be released either under the same or under a future GPL version, does the GPL also define who is eligible to define a new version of the GPL?
If you read the FSF's page on how to use the GPL for your software, the exact language they recommend for a GPLvN+ (GPL version N or any later) licence grant is
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
That makes it clear that only things called the GNU GPL as released by the FSF qualify for the optional upgrade. The language that makes that clear is in the the original software's licence grant, not the GPL itself.
It is certainly possible that someone could carelessly release software with a grant that said
you can redistribute it and/or modify it under the terms of the GNU General Public License, either version 3 of the License, or (at your option) any later version.
which is to say, a grant that failed to define whose later version of the GPL could be so used. But the GPL itself says that
Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
Moreover, the FSF have addressed the question of others using the GNU GPL as a basis for their own licence, and they say
You can legally use the GPL terms (possibly modified) in another license provided that you call your license by another name [...] and not mention GNU
So: say someone else were to muddy the waters by issuing something they called "the GNU GPL version 6" (which, for the sake of argument, allowed anyone with red hair to do anything they liked with all GPLv2+ and GPLv3+ content) the FSF would likely sue for copyright violation in short order. Following their probable success, courts would almost certainly take an extremely dim view of any red-haired person attempting to use the GPLvN+ licence grant, and "the GNU GPL version 6", to justify doing something that the GNU GPL would not iself permit.