GQ(4, 2)-extensions, strongly regular case

A strongly regular locallyGQ(4, 2)-graph is a graph with parameters either (126, 45, 12, 8) or (190, 45, 12, 10). The existence and the uniqueness of the corresponding locallyGQ(4, 2)-graph in the first case are well known. We prove that theGQ(4, 2)-hyperoval on ten vertices either is the Petersen graph, or is the Möbius 5-prism, or consists of two (2, 3)-subgraphs connected by three edges. We obtain homogeneousGQ(4, 2)-solutions with a strongly regular point graph; in particular, this implies the negative answer to the question of F. Buekenhout concerning the existence of a locallyGQ(4, 2)-graph with the parameters (190, 45, 12, 10).