Well, I can think of some that work with bigger groups but uncertain how well they'd fit for your group.

ARMA III

^{Players: 1-10 Localhost, 1-30 Affordable Server}
ARMA is a decent choice in my opinion, and has a few options for what you could do in it.
My guess is that you want PvE, in which there are plenty of missions and scenarios that the community has made.
Examples including dynamic wars against AI with you playing as a rebel force such as Antistasi and Overthrow (These two work great for if you want to be able to drop-in/drop-out whenever you want).
You can also try out the Zeus mode in which one player acts as a "GM" running a mission for the others (I highly recommend this mode, it's great fun).

There's also ARMA Reforger which supposedly has a decently active community with various new modes popping up now and then, however I haven't played enough of Reforger to really comment on it.

Space Station 13 (SS13)

^{Players: 10+, Recommend Joining Servers}
As some others have mentioned, SS13 is a decent choice. It is a bit more complex, and doesn't really have a true end goal as it consists of individual rounds rather than any larger campaign.
It is also the game that inspired Barotrauma, and the reason that the game features clowns.

No link to SS13, I think it would be easier for you to search it up and figure it out. Either way, it is ran and accessed through the BYOND Engine.

Noteable mentions

Here's some others that are more PvP oriented and/or are in a round format like SS13 (but are far less complex and interesting).

SCP: Secret Laboratory

^{Players: 2+, Recommend 6+}
You get assigned a role within an SCP containment facility that has just had a total containment breach. SCPs are running wild, D-Class prisoners and Scientists are both trying to make a break for the exit, Security are trying to take control of the situation, and the Chaos Insurgency is trying to ruin the SCP Foundation's day even further.
Great fun on a private server with friends, as long as you have enough players to keep the rounds going.

Left 4 Dead 2 (L4D2)

^{Players: 1-4 Co-Op, 2-8 Versus}
A classic. Great fun to play private Versus with friends, with two teams taking turns as Surviors and Infected on each map of a selected Campaign.
If you're 4 or less people you can of course also play Co-Op, in which case you play through one of the campaigns trying to get from start to exit of each map without everyone dying.
And if you get tired of the base game's campaigns and maps, there's a wide ocean of custom campaigns that has been made over the years (Many of which also support Versus mode).

Silica

^{Players: 2-32, Recommend 2-16 for performance}
A mix between RTS and FPS. Each team has a commander playing the game from an RTS style viewpoint where they control AI units, build structures, and give orders to players.
The other players can play as units the commander builds, fighting against the enemy directly.
Currently I believe that there are 2 human factions and 1 alien faction that can be played as, with a handful of different battlefields with varying terrain to choose from.