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The aim of the present paper is to expose two contributions of Mackey,together with a more recent result of Kawanaka and Matsuyama, generalized byBump and Ginzburg, on the representation theory of a finite group equipped withan involutory anti-automorphism (e.g. the anti-automorphism $g\mapsto g^{-1}$).Mackey's first contribution is a detailed version of the so-called Gelfandcriterion for weakly symmetric Gelfand pairs. Mackey's second contribution is acharacterization 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 afinite group $G$ and its automorphism $\sigma$ satisfy Condition ($\bigstar$)if the following condition is satisfied: if for $x,y\in G$, $x\cdotx^{-\sigma}$ and $y\cdot y^{-\sigma}$ are conjugate in $G$, then they areconjugate in $K=C_G(\sigma)$. We study the meanings of this condition, as wellas showing many examples of $G$ and $\sigma$ which do (or do not) satisfyCondition ($\bigstar$).

Mackey's theory of $\tau$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes