  The aim of the present paper is to expose two contributions of Mackey,
together with a more recent result of Kawanaka and Matsuyama, generalized by
Bump and Ginzburg, on the representation theory of a finite group equipped with
an involutory anti-automorphism (e.g. the anti-automorphism $g\mapsto g^{-1}$).
Mackey's first contribution is a detailed version of the so-called Gelfand
criterion for weakly symmetric Gelfand pairs. Mackey's second contribution is a
characterization of simply reducible groups (a notion introduced by Wigner).
The other result is a twisted version of the Frobenius-Schur theorem, where
"twisted" refers to the above-mentioned involutory anti-automorphism. APPENDIX:
We consider a special condition related to Gelfand pairs. Namely, we call a
finite group $G$ and its automorphism $\sigma$ satisfy Condition ($\bigstar$)
if the following condition is satisfied: if for $x,y\in G$, $x\cdot
x^{-\sigma}$ and $y\cdot y^{-\sigma}$ are conjugate in $G$, then they are
conjugate in $K=C_G(\sigma)$. We study the meanings of this condition, as well
as showing many examples of $G$ and $\sigma$ which do (or do not) satisfy
Condition ($\bigstar$).
