In 1896, Frobenius began the study of character theory while factoring the group determinant. Later in 1963, Brauer pointed out that the relationship between characters and their groups was still not fully understood. He published a series of questions that he felt would be important to resolve. In response to these questions, Johnson, Mattarei, and Sehgal developed the idea of a weak Cayley table map between groups. The set of all weak Cayley table maps from one group to itself also has a group structure, which we will call the weak Cayley table group. We will examine the weak Cayley table group of AGL(1; p) and the dicyclic groups, a nd a normal subgroup of the weak Cayley table group for a special case with Camina pairs and Semi-Direct products with a normal Hall-π subgroup, and look at some nontrivial weak Cayley table elements for certain p-groups. We also define a relative weak Cayley table and a relative weak Cayley table map. We will examine the relationship between relative weak Cayley table maps and weak Cayley table maps, automorphisms and anti-automorphisms, characters and spherical functions.