Frobenius Pairs with Perfect Involutions

An involution $i$ of a group $G$ is said to be perfect in $G$ if any two non-commuting involutions in $i^G$ are conjugated by an involution in the same class. We generalize theorems of Jordan and M. Hall concerning sharply doubly transitive groups, and the Shunkov theorem on periodic groups with a finite isolated subgroup of even order.