We consider the growth of an infinite family of finite groups. We are motivated by the remarkable contribution of Bass, Wolf, Milnor, Gromov, Grigorchuk on the word growth and structure of infinite groups, and the results of Black on the word growth of an infinite family of finite groups. We follow the definition of the word growth for families of finite groups as given by Black, and compute the growth of a family of finite linear fractional groups. Some developments are analogous to infinite cases. However, contrasts are also transcribed, as well as other results.