On rationality and 2-reflexiveness of wreath products of finite groups

A finite group $G$ is said to be rational if each its irreducible character acquires only rational values, and it is said to be 2-reflexive if each its element can be represented as a product of at most two involutions. We find necessary and sufficient conditions for the wreath of two finite groups be rational and 2-reflexive. Namely, we show that the wreath $H\wr K$ of two finite groups $H$ and $K$ is a rational (respectively, 2-reflexive) group iff $H$ is a rational (respectively, 2-reflexive) group and $K$ is an elementary Abelian 2-group. As a corollary, we obtain a description of all classical linear groups over finite fields of odd characteristic with rational and 2-reflexive Sylow 2-subgroups.