Final answer:
Extinct gnathostome groups include acanthodians, placoderms, dinosaurs, pterosaurs, ichthyosaurs, and morganucodonts. Acanthodians and placoderms are early fish groups that have been replaced by more modern fish and tetrapod lineages. Dinosaurs, pterosaurs, ichthyosaurs, and morganucodonts were also gnathostomes that no longer exist.
Step-by-step explanation:
Extinct Gnathostome Groups
Gnathostomes, or "jaw-mouths," include a wide array of vertebrates such as the cartilaginous fishes, the bony fishes, and all tetrapods. However, not all gnathostomes have survived into the present day. The question you've asked is which gnathostome groups are extinct. Two early groups of gnathostomes that are now extinct are the acanthodians and placoderms. These extinct gnathostomes arose in the late Silurian period and were replaced during the Devonian period due to the expansion of modern gnathostome groups which include Chondrichthyes (sharks, rays, skates, and ghost sharks) and Osteichthyes (bony fishes) such as the Actinopterygii (ray-finned fishes) and Sarcopterygii (lobe-finned fishes). The evolution of the jaw and paired fins were crucial developments that allowed these gnathostomes to become active predators and exploit new food sources.
Additionally, other extinct gnathostome groups from the Mesozoic period include the dinosaurs, pterosaurs, and ichthyosaurs, which were part of the diapsid classification. Within mammals, an extinct basal group to mention would be the morganucodonts, insectivorous creatures that lived during the late Triassic and early Jurassic periods.