Certain residual properties of HNN-extensions with central associated subgroups

Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form ${\prod }_{y\in Y}{X}_y$, where $X, Y \in $\mathcal{C}$$and$X_{y}$isanisomorphiccopyof$X$foreach$y \in Y$.