Linear representation theory of nontrivial semidirect product of Z4 and Z4

This article discusses the linear representation theory of the group nontrivial semidirect product of Z4 and Z4 (GAP ID: (16,4)), given by the presentation:

$$\langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$$

Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.