Fundamental groups of tangent conic-line arrangements with singularities up to order 6

We list all the possible fundamental groups of the complements of real conic-line arrangements with two conics which are tangent to each other at two points, with up to two additional lines. For the computations we use the topological local braid monodromies and the techniques of Moishezon–Teicher and van-Kampen. We also include some conjectures concerning the connection between the presentation of the fundamental group of the complements and the geometry of an interesting family of conic-line arrangements.