Involutions of the general linear group $GL_2$ over a subring of the field $\mathbb{Q}$

The necessary and sufficient conditions for the conjugacy of involutions of the group $GL_2$ over an arbitrary subring of the field $\mathbb{Q}$ are obtained. It is shown that if this subring is (not) $2$-divisible, then the number of classes of conjugate off-center involutions is $1$ (respectively $2$).