The General Services Administration, which last month began its second year operating under an acting administrator, appears beset these days by quarreling internal factions.

"It's like we have a bunch of children who can't play well together now," said a senior GSA official. "Instead of trying to solve problems, it seems that we try to try them in the press."

Dwight A. Ink, who has been acting GSA administrator since Ray A. Kline resigned last month, has tried repeatedly to quell the disputes. At one staff meeting last week, he chastized staffers for "airing dirty laundry" in public, according to participants. Ink said in a subsequent interview that he had never seen "anything like it" in his 30 years in government.

Over the past year, important Reagan administration management initiatives have run into a logjam at the GSA, particularly efforts to sell surplus federal property and to manage federal office space. For example, Earl E. Jones, commissioner of the GSA's Federal Property Resources Service, said last week that agencies have turned over so few properties for the GSA to sell that the GSA is no longer planning to expand its sales staff.

Meanwhile, plans to improve the procurement system and streamline the GSA's field structure have stalled because of the disputes and the uncertainty associated with an acting administrator.

The floundering, in the eyes of some, was epitomized by an order that Kline signed on his final day in office. Associate administrator Charles S. Davis said he interpreted the order as saying that the Federal Protective Service should move ahead on his idea of eliminating 350 of its mid-level supervisors. But Ink says that the memo said only that the recommendation would be studied further.

The rifts within the GSA extend to the public affairs office. Director Patrick H. McKelvey came in at the behest of Jack E. Courtemanche, the White House aide nominated as GSA administrator last year. But Courtemanche's nomination never cleared the Senate.

McKelvey reportedly has lost the confidence of many top GSA officials. Critics say he rarely consults career officials in the public affairs office. They also look askance at his decision to put Mark Quigley, a young, inexperienced political appointee, in charge of press affairs for the GSA's largest division, the Public Buildings Service.

According to aides, Ink has also been miffed by McKelvey's insistence on tape recording agency officials' interviews and then not advising other senior officials when they might want to contribute information.

McKelvey contended that Ink wants him to tape record conversations. But he added that Quigley, in several interviews "did not know that a GSA official was misstating a position or that it needed to be amplified."

Ink apparently was irritated when McKelvey characterized two consultants and a political appointee that Ink had hired as "just old friends not hired through competitive bidding." A senior aide to Ink said: "There are better ways to say it. The men brought aboard have short assignments, seven to 22 days, and the tab is less than $5,000." Ink acknowledged that the men are long-time acquaintances but said they are specialists who were brought on board to help him solve some short-term problems.

Ink's senior aides have urged reporters to rely on the administrator's inner circle rather than the public affairs office for help in interpreting events. "That's something that plagues this agency," McKelvey said. "There are people who are closer to Mr. Ink than I am."

McKelvey said he didn't expect to be "Golden's man," referring to Terence C. Golden, the assistant secretary of Treasury for administration, who has been nominated to be administrator. Golden's nomination is awaiting confirmation hearings before the Senate Governmental Affairs Committee.

"It's hard to say that we think we're floundering," said another of Ink's aides. "But things might work better if we had a full-time, regular administrator."