$\mathrm{C}\mathrm{S}\mathfrak{X}$-groups and free constructions

Let $\text{X}$ be a variety of groups. A group $G$ is $\text{CSX}$ if its maximal $\text{X}$-subgroups are malnormal. We prove that the amalgamated free product of two $\text{CSX}$-groups with a malnormal amalgam and without any involutions is again $\text{CSX}$. Using this, we describe conditions under which the HNN-extension of a $\text{CSX}$-group is $\text{CSX}$.