On realizations of the Gelfand character of a finite group

We show that the Gelfand character $ \chi_G$ of a finite group $G $ (i.e. the sum of all irreducible complex characters of $G$) may be realized as a "twisted trace" $ g \mapsto Tr(\rho_g \circ T) $ for a suitable involutive linear automorphism of $L^2(G)$, where $ \rho$ stands for the right regular representation of $G$ in $L^2(G)$. We prove further that, under certain hypotheses, $ T $ may be obtained as $T(f)= f \circ L, $ where $ L $ is an involutive antiautomorphism of the group $ G $ so that $ Tr(\rho_g \circ T) =|\{h \in G: hg = L(h) \}| $. We also give in the case of the group $G = PGL(2,\mathbb F_q) $ a positive answer to a question of K. W. Johnson asking whether it is possible to express the Gelfand character $ \chi_G$ as a polynomial in a single irreducible character $\eta$ of $G. $