External points to a conic from a Baer subplane

For an irreducible conic $\mathcal C$ in a Desarguesian plane of odd square order, estimating the number of points from a Baer subplane which are external to $\mathcal C$ is a natural problem. In this paper, a complete list of possibilities is determined for the case where $\mathcal C$ shares at least one point with the subplane.

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