I was looking at the description of the tetragonal group $$P 4_{2} 2 2$$ (No 93 in the International Tables of Crystallography) and there is one aspect that I do not understand. Namely, what is the convention and implications of choosing the origin point? For example, for the two fold rotation along the [110] plane, the origin is choosen to be $$(0, 0, \frac{1}{4})$$ and thus we should have the following transformation:
$$x \rightarrow y, y \rightarrow x, z \rightarrow -z + \frac{1}{2}$$.

Why the origin is not set at (0, 0, 0) and then one would have:
$$x \rightarrow y, y \rightarrow x, z \rightarrow -z$$?