Perform a fake Apollo-11 mission in a studio while exploring the base with a different team.
You have practically two missions going. One for the people (really being a Fake-Apollo on the ground/in a studio) and one to explore the alien base for real (that what we know as the "Apollo-11" mission). The people will "see" the rocket taking off and all, but the astronauts are swapped for a different crew shortly before start.
Planning the Conspiracy
To pull that off, you need to produce all the radio traffic and imagery of the landing in a studio. You need a great mind envisioning everything, direct and produce it prior to the mission (Previous Apollo flight recordings can help a great deal here). Then install the tape in the real Apollo-11 and let it playback at some point. This will produce all the radio talking (and video footage - if it was real time on the moon... idk atm) needed to convince the planet, that the mission went "smoothly" and that all the signals were really coming from space and were not an earth bound transmission only.
The Apollo astronauts (those who are publicly known) are then swapped by the Away-Team that will be the ones sitting in the real Saturn V to the moon. This crew is maybe more specialized in using weapons in space etc. and has suitable equipment. The guys are more like marines than astronauts etc.
This means, the designated Apollo astronauts need to stay on the planet. They are either real astronauts part of the conspiracy (could be hard for them to chew on, they were "The Right Stuff" gung ho guys... watch free on archive.org) or they were hand-picked "dummies"/actors from the beginning to play the roles of astronauts really went on a trip to the moon (Some hypnosis stuff might help here to imprint memories of the "never-happened"-mission).
Ground Control then has a secret radio channel (encrypted) which will provide the contact to the Away-Team going to explore that base.
After the Away-Team returns, the original astronauts/dummies are swapped back in again, before going public (could be done during the helicopter flight after salvaging the capsule; even if actually none or few of the Away-Team survived and returned).
Increasing team size
Beneficial would be to have some men extra in that crew. They could ride along in the command module, while others sit in the lander module (the lander needs to be adapted to support people sitting vertically during lift-off).
At least one pilot needs to stay in the command module for the mission.
Returning to earth
In the worst case nobody of the fake crew survives. The pilot then has to remotely ignite the start of the ascend station (upper part of the lander), dock it and return to earth (starting a new tape for the journey back), so the planet can witness the re-entry and all.
The ascend module needs to be ignited to be plausible that it was being used. Also to get the collected rocks etc. back into the command vehicle. Also some things need to be prepared for real by the Away-Team, putting up a flag, leaving lots of footprints, collect some rocks, put up some instruments etc., but this can be done quickly by one man let alone a larger team).
You can buy the Away-Team much time by for example adding extra rest phases or elongated sleeping-phases for the "exhausted pilots" (on tape). Meanwhile the Away-Team has slept during approach to the moon and starts its operation already.
If the Away-Team does survive the exploration, they can ride to earth Apollo-13 style in the lander and command module (although, this is future knowledge at that point... in our real Apollo-11, the ascend stage was jettisoned in lunar orbit before leaving towards earth). Re-entry with 5-6 astronauts will be a little crammed in the return capsule ("spam in a can"). Let's hope they planned for that.
Maybe other adjustments have to be considered too, like more oxygen for the crew etc.
Multiple Missions
I think after the initial Apollo-11 fake mission, the reality is so much altered that all future events are plot relevant and not world-building anymore. We do not know if the base was populated maybe and which artifacts could be salvaged there. In the alternate timeline they also can unveil the base to the public and justify returning there with the "truth" eventually. That is up to you.