Mums Kitchen Floor

There are a large number of possible answers here. Each one can easily be checked to see that it has the right symmetries.

For the extension, colour the squares like a chessboard. When you remove two opposite squares you remove two squares of the same colour. So you have left 30 squares of one colour and 32 of the other. You can’t cover these with the combination tiles as each combination covers one square of each colour.