Unique 2x2 Squares

1 1 1 1 0
1 1 0 0 1
0 1 0 0 1
0 0 0 1 0
1 1 0 1 1
  Is it possible to fill each square of a 5x5 grid with either a 1 or a 0, such that every 2x2 square is unique?  The example has one repeated square.

Source: Original.  Found later as a color-map question in Aaron J. Friedland's book Puzzles in Math & Logic (New York, 1970) on pages 9-10 and 44.

Solutions were received from MArk Rickert, Alan O'Donnell, Joseph DeVincentis, Philippe Fondanaiche, Yaacov Yoseph Weiss, Denis Borris, Charles Suprin, Kirk Bresniker, Claudio Baiocchi, Diane Fite, Oliver Stone, Bernie Erickson, Jean-Louis Legrand, David Madfes, Hidefumi Takahashi, Michael Shafer, and Muhammed Al-Humaid.

There are 800 solutions for the problem as stated.  Several solvers noticed the problem could actually be solved as a 4x4 grid, allowing for wrapping at the sides and top/bottom (equivalent to saying the first and last rows and columns above are the same.)  Neglecting rotations, reflections, 0-to-1 inversions, and cycles, there is only one such solution.  Yaacov Yoseph Weiss provided it first:


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