Grumman

Grumman is Colonel Roy Mustang's commanding officer and mentor at East City headquarters. He is first introduced playing chess with Roy Mustang when the transfer to Central has came for Roy. Mustang then bests the Lieutenant General in the game of chess (for the first time), and the General gives him the board along with the pieces secretly containing information as a farewell gift. Grumman is an old man that takes a lighthearted approach to his command. When Roy Mustang transfers from Eastern Headquarters to Central, Grumman authorizes the transfer of Mustang's subordinates, including his own granddaughter Riza Hawkeye (though it is unknown if she is aware of this relationship).

In the manga and 2009 anime
After the Homunculi strip Mustang of his subordinates, he calls for Grumman's aid. Grumman, not under the Homunculi's surveillance, acts on Mustang's behalf in intelligence gathering and communication, going as far as dressing up like an old woman to not be spotted in Central. Though a willing and powerful ally, he is reluctant to openly work against the military; as being branded a traitor would make it difficult to become the next leader of Amestris. It is implied in a joke that Grumman doesn't do anything in the East City Headquarters when he says to Roy: "Now that your transfer to Central has become final, I may have to start doing things around here again", although this may just be an inside joke between the two.

He next aids in the military coup, though he avoids having any sort of affiliations with Mustang or Olivier Armstrong. He does so because it would make it easier for him to take control of Amestris and become the next Fuhrer once the struggle is over. Grumman falls victim to the Nationwide Transmutation Circle, but is later revived thanks to Hohenheim's counter plan.

After the battle with Father is over, Roy passes the temporary rank of Führer over to Grumman. He allows Mrs. Bradley to keep Selim despite harsh opposition and wonders if humans and Homunculi can truly coexist. This makes him marvel at the fact that life can teach people so many things.

In the 2003 anime
Grumman's introduction is exactly the same as in the manga, but he plays no further importance to the plot after Roy's transfer to Central City, other than being present in meetings with the Fuhrer and commenting a few times.

He does make a brief cameo in Conqueror of Shamballa implying to have replaced King Bradley as the leader of Amestris.

Trivia

 * The kinship between Grumman and Lt. Hawkeye (Riza) is not declared at any point during the course of the manga, but it is stated at the Perfect Guidebook 2. The book reveals he's her grandfather on her mother's side (and therefore, don't bear the same surname). It's unclear whether Riza is aware of this kinship, since she states she doesn't know about any relatives (to Mustang, in Chapter 60 of the manga), but he's aware, as he even suggests Mustang to 'marry his granddaughter', so she would become the Fürher's wife.
 * Like nearly all military personnel in the Fullmetal Alchemist series, Gen. Grumman is named after a military vehicle, vessel or weapon, in his case, the Grumman E-2 Hawkeye, an aircraft carrier-based tactical Airborne Early Warning (AEW) aircraft used in the US Navy. Interesting enough, his granddaughter's name was also took from the same aircraft.
 * Grumman's record for playing chess with Mustang is 97 wins, 1 loss and 15 draws.