Blocking semiovals in PG(2,q2), q odd, admitting PGL(2,q) as an automorphism group

A (conic) blocking semioval is a set of points in a projective plane (containing a conic) that is both a blocking set and a semioval. Recently, Dover, Mellinger, and Wantz constructed two new families of conic blocking semiovals in the Desarguesian projective planes of odd order. One of their examples, say S, arises by considering in PG(2,q2), q odd, a unitary polarity commuting with an orthogonal polarity. In particular, such a conic blocking semioval has q3−q2+q2+1 points and is stabilized by a group G isomorphic to PGL(2,q). In this paper, we show that in the same geometric setting there are conic blocking semiovals, having the same size, admitting the same group G, but not isomorphic to S.