Unbounded classification of finite simply reducible groups

Classify all finite simply reducible groups without imposing an upper bound on their order.

Background

A finite simply reducible (SR) group is a finite group in which every element is conjugate to its inverse and every irreducible constituent of a tensor product of two irreducible representations occurs with multiplicity at most one. The paper gives a computer-assisted classification of nontrivial SR-groups only for orders at most 2000, including a separate construction for order 1024.

The authors explicitly identify the unrestricted classification as unresolved, while the present work addresses only the bounded range. The cited source labels this question as Problem 11.94.