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A finitely generated residually finite group $G$ is an $\widehat{OE}$-groupif any action of its profinite completion $\widehat G$ on a profinite tree withfinite edge stabilizers admits a global fixed point. In this paper, we studythe profinite genus of free products $G_1*_HG_2$ of $\widehat{OE}$-groups$G_1,G_2$ with finite amalgamation $H$. Given such $G_1,G_2,H$ we give preciseformulas for the number of isomorphism classes of $G_1*_HG_2$ and of itsprofinite completion. We compute the genus of $G_1*_HG_2$ and list varioussituations when the formula for the genus simplifies.

Profinite genus of free products with finite amalgamation