A commutativity criterion for a group of odd order

A finite group $G$ is called simply reducible ($SR$-group) if it has the following two properties: 1. Any element of this group is conjugate to its inverse. 2. The tensor product of any two irreducible representations is decomposed into a sum of irreducible representations of the group $G$ with multiplicities at most one. There are some generalizations of $SR$-groups. In particular, a finite group $G$ is called $ASR$-group if the tensor square of any irreducible representation $G$ is decomposed into a sum of irreducible representations of this group with multiplicities at most one. It has been proved that $ASR$-groups of odd order are abelian.