  A finitely generated residually finite group $G$ is an $\widehat{OE}$-group
if any action of its profinite completion $\widehat G$ on a profinite tree with
finite edge stabilizers admits a global fixed point. In this paper, we study
the 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 precise
formulas for the number of isomorphism classes of $G_1*_HG_2$ and of its
profinite completion. We compute the genus of $G_1*_HG_2$ and list various
situations when the formula for the genus simplifies.
