Catalysis of the reaction: 2 L-glutamate + 2 oxidized ferredoxin = L-glutamine + 2-oxoglutarate + 2 reduced ferredoxin + 2 H+. This is a two-step reaction: (a) L-glutamate + NH3 = L-glutamine + H2O, (b) L-glutamate + 2 oxidized ferredoxin + H2O = NH3 + 2-oxoglutarate + 2 reduced ferredoxin + 2 H+.