Complex Representations of Groups and Involutions of its Automorphisms

In this work, we establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. We develop algorithmic frameworks for evaluating these quantities in the context of inner automorphisms and the symmetric group ${\mathfrak{S}}_n$. As an application, we provide a criterion for identifying groups that possess complex (non-real) irreducible representations and explore the structural consequences arising from these results.