On the root-class residuality of certain free products of groups with normal amalgamated subgroups

Let $\mathcal{K}$ be a root class of groups closed under taking quotient groups, $G$ be a free product of groups $A$ and $B$ with amalgamated subgroups $H$ and $K$. Let also $H$ be normal in $A$, $K$ be normal in $B$, and $\operatorname{Aut}_{G}(H)$ denote the set of automorphisms of $H$ induced by all inner automorphisms of $G$. We prove a criterion for $G$ to be residually a $\mathcal{K}$-group provided $\operatorname{Aut}_{G}(H)$ is an abelian group or it satisfies some other conditions. We apply this result in the cases when $A$ and $B$ are bounded nilpotent groups or $A/H, B/K \in \mathcal{K}$.