This is quite interesting. Back in the 1920s, Lóránd Méray-Horváth, who set up the Méray Motorcycle company in Budapest in 1923 bought a Lafitte in France. His son described its friction drive to me. The Méray company apparently created a touring car prototype, based on the Lafitte. Though the photo is dated 1924, I believe this happened a bit later, around 1928.
As for the puzzle car, is that a 7 HP model?